27:
930:
913:
1573:
3038:
3539:
2776:
2921:
2502:
2194:
2335:
2648:
3176:
1918:
1453:
3310:
For an equation to be meaningful, the coefficient of at least one variable must be non-zero. If every variable has a zero coefficient, then, as mentioned for one variable, the equation is either
2076:
1458:
773:
125:
231:
2929:
1692:
293:
171:
1826:
1759:
1169:
1105:
3408:
546:
860:
821:
1331:
1385:
1625:
1261:
347:
1442:
609:
1003:
700:
2656:
507:
481:
3278:
372:
in the
Euclidean plane, and, conversely, every line can be viewed as the set of all solutions of a linear equation in two variables. This is the origin of the term
3251:
3298:
2784:
2365:
2084:
1839:
A line that is not parallel to an axis and does not pass through the origin cuts the axes into two different points. The intercept values
878:
is a function that maps a sum to the sum of the images of the summands. So, for this definition, the above function is linear only when
885:, that is when the line passes through the origin. To avoid confusion, the functions whose graph is an arbitrary line are often called
2209:
3307:-tuple such that substituting each element of the tuple for the corresponding variable transforms the equation into a true equality.
2525:
3080:
1568:{\displaystyle {\begin{aligned}m&=-{\frac {a}{b}},\\x_{0}&=-{\frac {c}{a}},\\y_{0}&=-{\frac {c}{b}}.\end{aligned}}}
3043:
Besides being very simple and mnemonic, this form has the advantage of being a special case of the more general equation of a
1863:
1178:
There are various ways of defining a line. In the following subsections, a linear equation of the line is given in each case.
3786:
3722:
3700:
2359:
This form is not symmetric in the two given points, but a symmetric form can be obtained by regrouping the constant terms:
2007:
723:
49:
3966:
3552:
3372:
if the coefficients are complex numbers or belong to any field). In the case of three variables, this hyperplane is a
1985:, there is exactly one line that passes through them. There are several ways to write a linear equation of this line.
3757:
3033:{\displaystyle {\begin{vmatrix}x&y&1\\x_{1}&y_{1}&1\\x_{2}&y_{2}&1\end{vmatrix}}=0.}
3961:
3945:
184:
20:
3534:{\displaystyle x_{j}=-{\frac {b}{a_{j}}}-\sum _{i\in \{1,\ldots ,n\},i\neq j}{\frac {a_{i}}{a_{j}}}x_{i}.}
4017:
3981:
3752:
3568:
1633:
252:
130:
1770:
1703:
1135:
1071:
3826:
643:
515:
430:
832:
249:, provided they do not contain any of the variables. To yield a meaningful equation, the coefficients
3779:
3747:
796:
1031:
The phrase "linear equation" takes its origin in this correspondence between lines and equations: a
313:
of such an equation are the values that, when substituted for the unknowns, make the equality true.
3971:
1285:
246:
1342:
3910:
2771:{\displaystyle {\begin{vmatrix}x-x_{1}&y-y_{1}\\x_{2}-x_{1}&y_{2}-y_{1}\end{vmatrix}}=0.}
1831:
for emphasizing that the slope of a line can be computed from the coordinates of any two points.
1590:
1221:
319:
1400:
567:
3940:
3935:
3930:
3808:
964:
779:
661:
174:
3675:
3920:
3900:
3655:
3343:
2200:
1009:
486:
448:
361:
178:
2356:(for verifying this, it suffices to verify that the two given points satisfy the equation).
1028:
are not both zero. Conversely, every line is the set of all solutions of a linear equation.
376:
for describing this type of equation. More generally, the solutions of a linear equation in
3986:
3905:
3772:
3549:
3256:
2923:
can be obtained by expanding with respect to its first row the determinant in the equation
425:
solutions and, more generally, to linear equations with coefficients and solutions in any
8:
3799:
3230:
1391:
1046:
791:
783:
426:
417:
This article considers the case of a single equation with coefficients from the field of
310:
303:
353:
refers implicitly to this particular case, in which the variable is sensibly called the
4022:
3976:
3843:
3838:
3764:
3573:
3283:
3074:
A linear equation with more than two variables may always be assumed to have the form
3821:
3718:
3711:
3696:
3583:
3059:
411:
299:
2507:(exchanging the two points changes the sign of the left-hand side of the equation).
3874:
3867:
3862:
3588:
3578:
3373:
3063:
2515:
The two-point form of the equation of a line can be expressed simply in terms of a
3996:
3693:
College
Mathematics for Business, Economics, Life Sciences and the Social Sciences
3884:
3879:
3816:
3668:
3648:
3365:
1266:
If, moreover, the line is not horizontal, it can be defined by its slope and its
1017:
1013:
887:
875:
787:
392:
369:
365:
3855:
3850:
871:
422:
402:
Linear equations occur frequently in all mathematics and their applications in
316:
In the case of just one variable, there is exactly one solution (provided that
4011:
3991:
3369:
421:, for which one studies the real solutions. All of its content applies to
3545:
2516:
1627:
of any point of the line. In this case, a linear equation of the line is
824:
418:
407:
238:
234:
35:
26:
3795:
3355:
3044:
2916:{\displaystyle (y_{1}-y_{2})x+(x_{2}-x_{1})y+(x_{1}y_{2}-x_{2}y_{1})=0}
2497:{\displaystyle (y_{1}-y_{2})x+(x_{2}-x_{1})y+(x_{1}y_{2}-x_{2}y_{1})=0}
1390:
These forms rely on the habit of considering a nonvertical line as the
900:
381:
298:
Alternatively, a linear equation can be obtained by equating to zero a
360:
In the case of two variables, each solution may be interpreted as the
3925:
242:
2189:{\displaystyle y-y_{1}={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}(x-x_{1}).}
3915:
867:
43:
1181:
1923:(It is easy to verify that the line defined by this equation has
1016:. With this interpretation, all solutions of the equation form a
403:
2330:{\displaystyle (x_{2}-x_{1})(y-y_{1})-(y_{2}-y_{1})(x-x_{1})=0,}
1857:
of these two points are nonzero, and an equation of the line is
2643:{\displaystyle (x_{2}-x_{1})(y-y_{1})-(y_{2}-y_{1})(x-x_{1})=0}
16:
Equation that does not involve powers or products of variables
3171:{\displaystyle a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}+b=0.}
1035:
in two variables is an equation whose solutions form a line.
429:. For the case of several simultaneous linear equations, see
2650:
is the result of expanding the determinant in the equation
1578:
929:
912:
862:
The functions whose graph is a line are generally called
3794:
1913:{\displaystyle {\frac {x}{x_{0}}}+{\frac {y}{y_{0}}}=1.}
2938:
2665:
1447:
these forms can be easily deduced from the relations
3666:
3646:
3411:
3286:
3259:
3233:
3083:
2932:
2787:
2659:
2528:
2368:
2212:
2087:
2010:
1866:
1773:
1706:
1636:
1593:
1456:
1403:
1345:
1288:
1224:
1138:
1074:
967:
835:
799:
726:
664:
570:
518:
489:
451:
322:
255:
187:
133:
52:
3695:(11th ed.), Upper Saddle River, N.J.: Pearson,
3690:
3607:
634:
are real numbers, it has infinitely many solutions.
3691:Barnett, R.A.; Ziegler, M.R.; Byleen, K.E. (2008),
3335:-tuples that are solutions of a linear equation in
2071:{\displaystyle {\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}.}
1053:that has been defined in the preceding section. If
3710:
3533:
3292:
3272:
3245:
3202:in old books). Depending on the context, the term
3170:
3032:
2915:
2770:
2642:
2496:
2329:
2188:
2070:
1912:
1820:
1753:
1686:
1619:
1567:
1436:
1379:
1325:
1255:
1163:
1099:
997:
854:
815:
768:{\displaystyle y=-{\frac {a}{b}}x-{\frac {c}{b}}.}
767:
694:
603:
540:
501:
475:
341:
287:
225:
165:
120:{\displaystyle a_{1}x_{1}+\ldots +a_{n}x_{n}+b=0,}
119:
414:are often well approximated by linear equations.
4009:
1583:A non-vertical line can be defined by its slope
1186:A non-vertical line can be defined by its slope
1182:Slope–intercept form or Gradient-intercept form
3708:
3619:
30:Two graphs of linear equations in two variables
3817:Zero polynomial (degree undefined or −1 or −∞)
3630:
3628:
3780:
3730:
3634:
3477:
3459:
1279:. In this case, its equation can be written
705:is a linear equation in the single variable
368:. The solutions of a linear equation form a
3650:An Elementary Course in Theory of Equations
3625:
3058:. These equations rely on the condition of
907:
3787:
3773:
3670:Numbers Universalized: An Advanced Algebra
3266:
3069:
1121:, the line is the graph of a function of
713:. It has therefore a unique solution for
306:, from which the coefficients are taken.
1207:coordinate of its intersection with the
1132:, one has a horizontal line of equation
1107:which is not the graph of a function of
928:
911:
241:. The coefficients may be considered as
25:
3709:Larson, Ron; Hostetler, Robert (2007),
1579:Point–slope form or Point-gradient form
4010:
3391:, then the equation can be solved for
2519:. There are two common ways for that.
3768:
3673:. American Book Company. p. 113.
1173:
891:, and the linear functions such that
245:of the equation and may be arbitrary
226:{\displaystyle b,a_{1},\ldots ,a_{n}}
3303:A solution of such an equation is a
3731:Wilson, W.A.; Tracey, J.I. (1925),
3379:If a linear equation is given with
2510:
1687:{\displaystyle y=y_{1}+m(x-x_{1}),}
1394:. For a line given by an equation
556:A linear equation in two variables
288:{\displaystyle a_{1},\ldots ,a_{n}}
166:{\displaystyle x_{1},\ldots ,x_{n}}
13:
3653:. J. Wiley & sons. p. 17.
3608:Barnett, Ziegler & Byleen 2008
1821:{\displaystyle y-y_{1}=m(x-x_{1})}
1764:This equation can also be written
1754:{\displaystyle y=mx+y_{1}-mx_{1}.}
1164:{\displaystyle y=-{\frac {c}{b}}.}
1100:{\displaystyle x=-{\frac {c}{a}},}
637:
441:A linear equation in one variable
14:
4034:
3740:
1944:
1834:
541:{\displaystyle x=-{\frac {b}{a}}}
295:are required to not all be zero.
3321:) as having no solution, or all
1064:(that is a line parallel to the
855:{\displaystyle -{\frac {c}{b}}.}
551:
3253:variables, it is common to use
3051:points in a space of dimension
816:{\displaystyle -{\frac {a}{b}}}
436:
3735:(revised ed.), D.C. Heath
3667:David Martin Sensenig (1890).
3660:
3647:Charles Hiram Chapman (1892).
3640:
3613:
3601:
3300:instead of indexed variables.
2904:
2858:
2849:
2823:
2814:
2788:
2631:
2612:
2609:
2583:
2577:
2558:
2555:
2529:
2485:
2439:
2430:
2404:
2395:
2369:
2315:
2296:
2293:
2267:
2261:
2242:
2239:
2213:
2180:
2161:
1815:
1796:
1678:
1659:
1317:
1298:
1:
3977:Horner's method of evaluation
3684:
1326:{\displaystyle y=m(x-x_{0}),}
3713:Precalculus:A Concise Course
2078:Thus, a point-slope form is
1380:{\displaystyle y=mx-mx_{0}.}
933:Horizontal line of equation
46:that may be put in the form
21:Linear differential equation
7:
3982:Polynomial identity testing
3753:Encyclopedia of Mathematics
3620:Larson & Hostetler 2007
3569:Linear equation over a ring
3562:
2004:, the slope of the line is
1949:Given two different points
1620:{\displaystyle x_{1},y_{1}}
1256:{\displaystyle y=mx+y_{0}.}
342:{\displaystyle a_{1}\neq 0}
10:
4039:
1437:{\displaystyle ax+by+c=0,}
1211:-axis). In this case, its
916:Vertical line of equation
644:Linear function (calculus)
641:
604:{\displaystyle ax+by+c=0,}
431:system of linear equations
18:
3954:
3893:
3806:
2340:which is valid also when
2203:, one gets the equation
998:{\displaystyle ax+by+c=0}
695:{\displaystyle ax+by+c=0}
384:(a subspace of dimension
3635:Wilson & Tracey 1925
3594:
3544:If the coefficients are
3206:can be reserved for the
908:Geometric interpretation
19:Not to be confused with
3967:Greatest common divisor
3070:More than two variables
502:{\displaystyle a\neq 0}
476:{\displaystyle ax+b=0,}
3839:Quadratic function (2)
3535:
3294:
3274:
3247:
3172:
3034:
2917:
2772:
2644:
2498:
2331:
2190:
2072:
1941:as intercept values).
1914:
1822:
1755:
1688:
1621:
1587:, and the coordinates
1569:
1438:
1381:
1327:
1257:
1165:
1101:
999:
943:
926:
856:
817:
786:of this function is a
769:
696:
605:
542:
503:
477:
343:
289:
227:
167:
121:
31:
3822:Constant function (0)
3536:
3344:Cartesian coordinates
3295:
3275:
3273:{\displaystyle x,\;y}
3248:
3173:
3035:
2918:
2773:
2645:
2499:
2332:
2201:clearing denominators
2191:
2073:
1915:
1823:
1756:
1689:
1622:
1570:
1439:
1382:
1328:
1258:
1166:
1102:
1047:graph of the function
1010:Cartesian coordinates
1008:may be viewed as the
1000:
958:of a linear equation
932:
915:
857:
818:
770:
717:, which is given by
697:
606:
543:
504:
478:
362:Cartesian coordinates
344:
290:
228:
168:
122:
29:
3955:Tools and algorithms
3875:Quintic function (5)
3863:Quartic function (4)
3800:polynomial functions
3717:, Houghton Mifflin,
3409:
3346:of the points of an
3284:
3257:
3231:
3081:
2930:
2785:
2657:
2526:
2366:
2210:
2085:
2008:
1864:
1771:
1704:
1634:
1591:
1454:
1401:
1343:
1286:
1222:
1136:
1072:
965:
833:
797:
724:
662:
568:
516:
487:
449:
320:
253:
185:
131:
50:
3885:Septic equation (7)
3880:Sextic equation (6)
3827:Linear function (1)
3676:Extract of page 113
3246:{\displaystyle n=3}
1392:graph of a function
1068:-axis) of equation
709:for every value of
349:). Often, the term
4018:Elementary algebra
3851:Cubic function (3)
3844:Quadratic equation
3656:Extract of page 17
3589:Nonlinear equation
3574:Algebraic equation
3531:
3493:
3290:
3270:
3243:
3227:When dealing with
3168:
3030:
3018:
2913:
2768:
2756:
2640:
2494:
2327:
2186:
2068:
1910:
1818:
1751:
1684:
1617:
1565:
1563:
1434:
1377:
1336:or, equivalently,
1323:
1253:
1174:Equation of a line
1161:
1097:
1045:, the line is the
1012:of a point in the
995:
944:
927:
866:in the context of
852:
813:
765:
692:
601:
564:can be written as
538:
499:
473:
445:can be written as
412:non-linear systems
364:of a point of the
339:
285:
237:, which are often
223:
163:
117:
32:
4005:
4004:
3946:Quasi-homogeneous
3748:"Linear equation"
3733:Analytic Geometry
3724:978-0-618-62719-6
3702:978-0-13-157225-6
3584:Linear inequality
3548:, this defines a
3516:
3448:
3443:
3293:{\displaystyle z}
3060:linear dependence
2159:
2063:
1902:
1882:
1556:
1519:
1482:
1156:
1092:
898:are often called
847:
811:
760:
744:
536:
410:, partly because
380:variables form a
300:linear polynomial
4030:
3868:Quartic equation
3789:
3782:
3775:
3766:
3765:
3761:
3736:
3727:
3716:
3705:
3678:
3674:
3664:
3658:
3654:
3644:
3638:
3632:
3623:
3617:
3611:
3605:
3579:Line coordinates
3556:
3540:
3538:
3537:
3532:
3527:
3526:
3517:
3515:
3514:
3505:
3504:
3495:
3492:
3444:
3442:
3441:
3429:
3421:
3420:
3401:
3390:
3364:
3362:
3353:
3341:
3339:
3334:
3327:
3325:
3320:
3306:
3299:
3297:
3296:
3291:
3279:
3277:
3276:
3271:
3252:
3250:
3249:
3244:
3223:
3216:
3193:
3185:, often denoted
3184:
3181:The coefficient
3177:
3175:
3174:
3169:
3155:
3154:
3145:
3144:
3126:
3125:
3116:
3115:
3103:
3102:
3093:
3092:
3064:projective space
3057:
3050:
3047:passing through
3039:
3037:
3036:
3031:
3023:
3022:
3010:
3009:
2998:
2997:
2979:
2978:
2967:
2966:
2922:
2920:
2919:
2914:
2903:
2902:
2893:
2892:
2880:
2879:
2870:
2869:
2848:
2847:
2835:
2834:
2813:
2812:
2800:
2799:
2777:
2775:
2774:
2769:
2761:
2760:
2753:
2752:
2740:
2739:
2728:
2727:
2715:
2714:
2701:
2700:
2683:
2682:
2649:
2647:
2646:
2641:
2630:
2629:
2608:
2607:
2595:
2594:
2576:
2575:
2554:
2553:
2541:
2540:
2511:Determinant form
2503:
2501:
2500:
2495:
2484:
2483:
2474:
2473:
2461:
2460:
2451:
2450:
2429:
2428:
2416:
2415:
2394:
2393:
2381:
2380:
2355:
2336:
2334:
2333:
2328:
2314:
2313:
2292:
2291:
2279:
2278:
2260:
2259:
2238:
2237:
2225:
2224:
2195:
2193:
2192:
2187:
2179:
2178:
2160:
2158:
2157:
2156:
2144:
2143:
2133:
2132:
2131:
2119:
2118:
2108:
2103:
2102:
2077:
2075:
2074:
2069:
2064:
2062:
2061:
2060:
2048:
2047:
2037:
2036:
2035:
2023:
2022:
2012:
2003:
1984:
1966:
1940:
1931:
1919:
1917:
1916:
1911:
1903:
1901:
1900:
1888:
1883:
1881:
1880:
1868:
1856:
1847:
1827:
1825:
1824:
1819:
1814:
1813:
1789:
1788:
1760:
1758:
1757:
1752:
1747:
1746:
1731:
1730:
1693:
1691:
1690:
1685:
1677:
1676:
1652:
1651:
1626:
1624:
1623:
1618:
1616:
1615:
1603:
1602:
1586:
1574:
1572:
1571:
1566:
1564:
1557:
1549:
1537:
1536:
1520:
1512:
1500:
1499:
1483:
1475:
1443:
1441:
1440:
1435:
1386:
1384:
1383:
1378:
1373:
1372:
1332:
1330:
1329:
1324:
1316:
1315:
1278:
1269:
1262:
1260:
1259:
1254:
1249:
1248:
1215:can be written
1210:
1206:
1202:
1193:
1189:
1170:
1168:
1167:
1162:
1157:
1149:
1131:
1124:
1120:
1110:
1106:
1104:
1103:
1098:
1093:
1085:
1067:
1060:, the line is a
1059:
1052:
1044:
1027:
1023:
1020:, provided that
1004:
1002:
1001:
996:
957:
942:
925:
897:
888:affine functions
884:
864:linear functions
861:
859:
858:
853:
848:
840:
827:
822:
820:
819:
814:
812:
804:
774:
772:
771:
766:
761:
753:
745:
737:
716:
712:
708:
701:
699:
698:
693:
655:, the equation
654:
633:
629:
622:
618:
614:
610:
608:
607:
602:
563:
559:
547:
545:
544:
539:
537:
529:
512:The solution is
508:
506:
505:
500:
482:
480:
479:
474:
444:
398:
390:
379:
348:
346:
345:
340:
332:
331:
294:
292:
291:
286:
284:
283:
265:
264:
232:
230:
229:
224:
222:
221:
203:
202:
172:
170:
169:
164:
162:
161:
143:
142:
126:
124:
123:
118:
101:
100:
91:
90:
72:
71:
62:
61:
4038:
4037:
4033:
4032:
4031:
4029:
4028:
4027:
4008:
4007:
4006:
4001:
3950:
3889:
3832:Linear equation
3802:
3793:
3746:
3743:
3725:
3703:
3687:
3682:
3681:
3665:
3661:
3645:
3641:
3633:
3626:
3618:
3614:
3606:
3602:
3597:
3565:
3554:
3522:
3518:
3510:
3506:
3500:
3496:
3494:
3452:
3437:
3433:
3428:
3416:
3412:
3410:
3407:
3406:
3400:
3392:
3388:
3380:
3366:Euclidean space
3360:
3359:
3347:
3337:
3336:
3332:
3328:are solutions.
3323:
3322:
3315:
3304:
3285:
3282:
3281:
3258:
3255:
3254:
3232:
3229:
3228:
3218:
3215:
3207:
3198:(sometimes the
3192:
3186:
3182:
3150:
3146:
3140:
3136:
3121:
3117:
3111:
3107:
3098:
3094:
3088:
3084:
3082:
3079:
3078:
3072:
3062:of points in a
3052:
3048:
3017:
3016:
3011:
3005:
3001:
2999:
2993:
2989:
2986:
2985:
2980:
2974:
2970:
2968:
2962:
2958:
2955:
2954:
2949:
2944:
2934:
2933:
2931:
2928:
2927:
2898:
2894:
2888:
2884:
2875:
2871:
2865:
2861:
2843:
2839:
2830:
2826:
2808:
2804:
2795:
2791:
2786:
2783:
2782:
2755:
2754:
2748:
2744:
2735:
2731:
2729:
2723:
2719:
2710:
2706:
2703:
2702:
2696:
2692:
2684:
2678:
2674:
2661:
2660:
2658:
2655:
2654:
2625:
2621:
2603:
2599:
2590:
2586:
2571:
2567:
2549:
2545:
2536:
2532:
2527:
2524:
2523:
2513:
2479:
2475:
2469:
2465:
2456:
2452:
2446:
2442:
2424:
2420:
2411:
2407:
2389:
2385:
2376:
2372:
2367:
2364:
2363:
2354:
2347:
2341:
2309:
2305:
2287:
2283:
2274:
2270:
2255:
2251:
2233:
2229:
2220:
2216:
2211:
2208:
2207:
2174:
2170:
2152:
2148:
2139:
2135:
2134:
2127:
2123:
2114:
2110:
2109:
2107:
2098:
2094:
2086:
2083:
2082:
2056:
2052:
2043:
2039:
2038:
2031:
2027:
2018:
2014:
2013:
2011:
2009:
2006:
2005:
2002:
1995:
1989:
1982:
1975:
1968:
1964:
1957:
1950:
1947:
1939:
1933:
1930:
1924:
1896:
1892:
1887:
1876:
1872:
1867:
1865:
1862:
1861:
1855:
1849:
1846:
1840:
1837:
1809:
1805:
1784:
1780:
1772:
1769:
1768:
1742:
1738:
1726:
1722:
1705:
1702:
1701:
1672:
1668:
1647:
1643:
1635:
1632:
1631:
1611:
1607:
1598:
1594:
1592:
1589:
1588:
1584:
1581:
1562:
1561:
1548:
1538:
1532:
1528:
1525:
1524:
1511:
1501:
1495:
1491:
1488:
1487:
1474:
1464:
1457:
1455:
1452:
1451:
1402:
1399:
1398:
1368:
1364:
1344:
1341:
1340:
1311:
1307:
1287:
1284:
1283:
1277:
1271:
1267:
1244:
1240:
1223:
1220:
1219:
1213:linear equation
1208:
1204:
1201:
1195:
1191:
1187:
1184:
1176:
1148:
1137:
1134:
1133:
1126:
1122:
1115:
1108:
1084:
1073:
1070:
1069:
1065:
1054:
1050:
1039:
1033:linear equation
1025:
1021:
1014:Euclidean plane
966:
963:
962:
947:
934:
917:
910:
892:
879:
876:linear function
839:
834:
831:
830:
825:
803:
798:
795:
794:
778:This defines a
752:
736:
725:
722:
721:
714:
710:
706:
663:
660:
659:
649:
646:
640:
638:Linear function
631:
627:
620:
616:
612:
569:
566:
565:
561:
557:
554:
528:
517:
514:
513:
488:
485:
484:
450:
447:
446:
442:
439:
396:
393:Euclidean space
385:
377:
366:Euclidean plane
351:linear equation
327:
323:
321:
318:
317:
279:
275:
260:
256:
254:
251:
250:
217:
213:
198:
194:
186:
183:
182:
157:
153:
138:
134:
132:
129:
128:
96:
92:
86:
82:
67:
63:
57:
53:
51:
48:
47:
40:linear equation
24:
17:
12:
11:
5:
4036:
4026:
4025:
4020:
4003:
4002:
4000:
3999:
3994:
3989:
3984:
3979:
3974:
3969:
3964:
3958:
3956:
3952:
3951:
3949:
3948:
3943:
3938:
3933:
3928:
3923:
3918:
3913:
3908:
3903:
3897:
3895:
3891:
3890:
3888:
3887:
3882:
3877:
3872:
3871:
3870:
3860:
3859:
3858:
3856:Cubic equation
3848:
3847:
3846:
3836:
3835:
3834:
3824:
3819:
3813:
3811:
3804:
3803:
3792:
3791:
3784:
3777:
3769:
3763:
3762:
3742:
3741:External links
3739:
3738:
3737:
3728:
3723:
3706:
3701:
3686:
3683:
3680:
3679:
3659:
3639:
3624:
3612:
3599:
3598:
3596:
3593:
3592:
3591:
3586:
3581:
3576:
3571:
3564:
3561:
3557:real variables
3542:
3541:
3530:
3525:
3521:
3513:
3509:
3503:
3499:
3491:
3488:
3485:
3482:
3479:
3476:
3473:
3470:
3467:
3464:
3461:
3458:
3455:
3451:
3447:
3440:
3436:
3432:
3427:
3424:
3419:
3415:
3396:
3384:
3289:
3269:
3265:
3262:
3242:
3239:
3236:
3211:
3194:is called the
3190:
3179:
3178:
3167:
3164:
3161:
3158:
3153:
3149:
3143:
3139:
3135:
3132:
3129:
3124:
3120:
3114:
3110:
3106:
3101:
3097:
3091:
3087:
3071:
3068:
3041:
3040:
3029:
3026:
3021:
3015:
3012:
3008:
3004:
3000:
2996:
2992:
2988:
2987:
2984:
2981:
2977:
2973:
2969:
2965:
2961:
2957:
2956:
2953:
2950:
2948:
2945:
2943:
2940:
2939:
2937:
2912:
2909:
2906:
2901:
2897:
2891:
2887:
2883:
2878:
2874:
2868:
2864:
2860:
2857:
2854:
2851:
2846:
2842:
2838:
2833:
2829:
2825:
2822:
2819:
2816:
2811:
2807:
2803:
2798:
2794:
2790:
2779:
2778:
2767:
2764:
2759:
2751:
2747:
2743:
2738:
2734:
2730:
2726:
2722:
2718:
2713:
2709:
2705:
2704:
2699:
2695:
2691:
2688:
2685:
2681:
2677:
2673:
2670:
2667:
2666:
2664:
2639:
2636:
2633:
2628:
2624:
2620:
2617:
2614:
2611:
2606:
2602:
2598:
2593:
2589:
2585:
2582:
2579:
2574:
2570:
2566:
2563:
2560:
2557:
2552:
2548:
2544:
2539:
2535:
2531:
2512:
2509:
2505:
2504:
2493:
2490:
2487:
2482:
2478:
2472:
2468:
2464:
2459:
2455:
2449:
2445:
2441:
2438:
2435:
2432:
2427:
2423:
2419:
2414:
2410:
2406:
2403:
2400:
2397:
2392:
2388:
2384:
2379:
2375:
2371:
2352:
2345:
2338:
2337:
2326:
2323:
2320:
2317:
2312:
2308:
2304:
2301:
2298:
2295:
2290:
2286:
2282:
2277:
2273:
2269:
2266:
2263:
2258:
2254:
2250:
2247:
2244:
2241:
2236:
2232:
2228:
2223:
2219:
2215:
2197:
2196:
2185:
2182:
2177:
2173:
2169:
2166:
2163:
2155:
2151:
2147:
2142:
2138:
2130:
2126:
2122:
2117:
2113:
2106:
2101:
2097:
2093:
2090:
2067:
2059:
2055:
2051:
2046:
2042:
2034:
2030:
2026:
2021:
2017:
2000:
1993:
1980:
1973:
1962:
1955:
1946:
1945:Two-point form
1943:
1937:
1928:
1921:
1920:
1909:
1906:
1899:
1895:
1891:
1886:
1879:
1875:
1871:
1853:
1844:
1836:
1835:Intercept form
1833:
1829:
1828:
1817:
1812:
1808:
1804:
1801:
1798:
1795:
1792:
1787:
1783:
1779:
1776:
1762:
1761:
1750:
1745:
1741:
1737:
1734:
1729:
1725:
1721:
1718:
1715:
1712:
1709:
1695:
1694:
1683:
1680:
1675:
1671:
1667:
1664:
1661:
1658:
1655:
1650:
1646:
1642:
1639:
1614:
1610:
1606:
1601:
1597:
1580:
1577:
1576:
1575:
1560:
1555:
1552:
1547:
1544:
1541:
1539:
1535:
1531:
1527:
1526:
1523:
1518:
1515:
1510:
1507:
1504:
1502:
1498:
1494:
1490:
1489:
1486:
1481:
1478:
1473:
1470:
1467:
1465:
1463:
1460:
1459:
1445:
1444:
1433:
1430:
1427:
1424:
1421:
1418:
1415:
1412:
1409:
1406:
1388:
1387:
1376:
1371:
1367:
1363:
1360:
1357:
1354:
1351:
1348:
1334:
1333:
1322:
1319:
1314:
1310:
1306:
1303:
1300:
1297:
1294:
1291:
1275:
1264:
1263:
1252:
1247:
1243:
1239:
1236:
1233:
1230:
1227:
1199:
1183:
1180:
1175:
1172:
1160:
1155:
1152:
1147:
1144:
1141:
1114:Similarly, if
1096:
1091:
1088:
1083:
1080:
1077:
1006:
1005:
994:
991:
988:
985:
982:
979:
976:
973:
970:
946:Each solution
909:
906:
872:linear algebra
870:. However, in
851:
846:
843:
838:
810:
807:
802:
776:
775:
764:
759:
756:
751:
748:
743:
740:
735:
732:
729:
703:
702:
691:
688:
685:
682:
679:
676:
673:
670:
667:
642:Main article:
639:
636:
600:
597:
594:
591:
588:
585:
582:
579:
576:
573:
553:
550:
535:
532:
527:
524:
521:
498:
495:
492:
472:
469:
466:
463:
460:
457:
454:
438:
435:
338:
335:
330:
326:
282:
278:
274:
271:
268:
263:
259:
220:
216:
212:
209:
206:
201:
197:
193:
190:
160:
156:
152:
149:
146:
141:
137:
116:
113:
110:
107:
104:
99:
95:
89:
85:
81:
78:
75:
70:
66:
60:
56:
15:
9:
6:
4:
3:
2:
4035:
4024:
4021:
4019:
4016:
4015:
4013:
3998:
3997:Gröbner basis
3995:
3993:
3990:
3988:
3985:
3983:
3980:
3978:
3975:
3973:
3970:
3968:
3965:
3963:
3962:Factorization
3960:
3959:
3957:
3953:
3947:
3944:
3942:
3939:
3937:
3934:
3932:
3929:
3927:
3924:
3922:
3919:
3917:
3914:
3912:
3909:
3907:
3904:
3902:
3899:
3898:
3896:
3894:By properties
3892:
3886:
3883:
3881:
3878:
3876:
3873:
3869:
3866:
3865:
3864:
3861:
3857:
3854:
3853:
3852:
3849:
3845:
3842:
3841:
3840:
3837:
3833:
3830:
3829:
3828:
3825:
3823:
3820:
3818:
3815:
3814:
3812:
3810:
3805:
3801:
3797:
3790:
3785:
3783:
3778:
3776:
3771:
3770:
3767:
3759:
3755:
3754:
3749:
3745:
3744:
3734:
3729:
3726:
3720:
3715:
3714:
3707:
3704:
3698:
3694:
3689:
3688:
3677:
3672:
3671:
3663:
3657:
3652:
3651:
3643:
3636:
3631:
3629:
3621:
3616:
3609:
3604:
3600:
3590:
3587:
3585:
3582:
3580:
3577:
3575:
3572:
3570:
3567:
3566:
3560:
3558:
3551:
3547:
3528:
3523:
3519:
3511:
3507:
3501:
3497:
3489:
3486:
3483:
3480:
3474:
3471:
3468:
3465:
3462:
3456:
3453:
3449:
3445:
3438:
3434:
3430:
3425:
3422:
3417:
3413:
3405:
3404:
3403:
3399:
3395:
3387:
3383:
3377:
3375:
3371:
3367:
3357:
3354:-dimensional
3351:
3345:
3329:
3318:
3313:
3308:
3301:
3287:
3267:
3263:
3260:
3240:
3237:
3234:
3225:
3221:
3214:
3210:
3205:
3201:
3200:absolute term
3197:
3196:constant term
3189:
3165:
3162:
3159:
3156:
3151:
3147:
3141:
3137:
3133:
3130:
3127:
3122:
3118:
3112:
3108:
3104:
3099:
3095:
3089:
3085:
3077:
3076:
3075:
3067:
3065:
3061:
3055:
3046:
3027:
3024:
3019:
3013:
3006:
3002:
2994:
2990:
2982:
2975:
2971:
2963:
2959:
2951:
2946:
2941:
2935:
2926:
2925:
2924:
2910:
2907:
2899:
2895:
2889:
2885:
2881:
2876:
2872:
2866:
2862:
2855:
2852:
2844:
2840:
2836:
2831:
2827:
2820:
2817:
2809:
2805:
2801:
2796:
2792:
2781:The equation
2765:
2762:
2757:
2749:
2745:
2741:
2736:
2732:
2724:
2720:
2716:
2711:
2707:
2697:
2693:
2689:
2686:
2679:
2675:
2671:
2668:
2662:
2653:
2652:
2651:
2637:
2634:
2626:
2622:
2618:
2615:
2604:
2600:
2596:
2591:
2587:
2580:
2572:
2568:
2564:
2561:
2550:
2546:
2542:
2537:
2533:
2522:The equation
2520:
2518:
2508:
2491:
2488:
2480:
2476:
2470:
2466:
2462:
2457:
2453:
2447:
2443:
2436:
2433:
2425:
2421:
2417:
2412:
2408:
2401:
2398:
2390:
2386:
2382:
2377:
2373:
2362:
2361:
2360:
2357:
2351:
2344:
2324:
2321:
2318:
2310:
2306:
2302:
2299:
2288:
2284:
2280:
2275:
2271:
2264:
2256:
2252:
2248:
2245:
2234:
2230:
2226:
2221:
2217:
2206:
2205:
2204:
2202:
2183:
2175:
2171:
2167:
2164:
2153:
2149:
2145:
2140:
2136:
2128:
2124:
2120:
2115:
2111:
2104:
2099:
2095:
2091:
2088:
2081:
2080:
2079:
2065:
2057:
2053:
2049:
2044:
2040:
2032:
2028:
2024:
2019:
2015:
1999:
1992:
1986:
1979:
1972:
1961:
1954:
1942:
1936:
1927:
1907:
1904:
1897:
1893:
1889:
1884:
1877:
1873:
1869:
1860:
1859:
1858:
1852:
1843:
1832:
1810:
1806:
1802:
1799:
1793:
1790:
1785:
1781:
1777:
1774:
1767:
1766:
1765:
1748:
1743:
1739:
1735:
1732:
1727:
1723:
1719:
1716:
1713:
1710:
1707:
1700:
1699:
1698:
1681:
1673:
1669:
1665:
1662:
1656:
1653:
1648:
1644:
1640:
1637:
1630:
1629:
1628:
1612:
1608:
1604:
1599:
1595:
1558:
1553:
1550:
1545:
1542:
1540:
1533:
1529:
1521:
1516:
1513:
1508:
1505:
1503:
1496:
1492:
1484:
1479:
1476:
1471:
1468:
1466:
1461:
1450:
1449:
1448:
1431:
1428:
1425:
1422:
1419:
1416:
1413:
1410:
1407:
1404:
1397:
1396:
1395:
1393:
1374:
1369:
1365:
1361:
1358:
1355:
1352:
1349:
1346:
1339:
1338:
1337:
1320:
1312:
1308:
1304:
1301:
1295:
1292:
1289:
1282:
1281:
1280:
1274:
1250:
1245:
1241:
1237:
1234:
1231:
1228:
1225:
1218:
1217:
1216:
1214:
1198:
1179:
1171:
1158:
1153:
1150:
1145:
1142:
1139:
1129:
1118:
1112:
1094:
1089:
1086:
1081:
1078:
1075:
1063:
1062:vertical line
1057:
1048:
1042:
1036:
1034:
1029:
1019:
1015:
1011:
992:
989:
986:
983:
980:
977:
974:
971:
968:
961:
960:
959:
955:
951:
941:
937:
931:
924:
920:
914:
905:
903:
902:
895:
890:
889:
882:
877:
873:
869:
865:
849:
844:
841:
836:
829:
808:
805:
800:
793:
789:
785:
781:
762:
757:
754:
749:
746:
741:
738:
733:
730:
727:
720:
719:
718:
689:
686:
683:
680:
677:
674:
671:
668:
665:
658:
657:
656:
652:
645:
635:
624:
619:are not both
598:
595:
592:
589:
586:
583:
580:
577:
574:
571:
552:Two variables
549:
533:
530:
525:
522:
519:
510:
496:
493:
490:
470:
467:
464:
461:
458:
455:
452:
434:
432:
428:
424:
420:
415:
413:
409:
405:
400:
395:of dimension
394:
388:
383:
375:
371:
367:
363:
358:
356:
352:
336:
333:
328:
324:
314:
312:
307:
305:
301:
296:
280:
276:
272:
269:
266:
261:
257:
248:
244:
240:
236:
218:
214:
210:
207:
204:
199:
195:
191:
188:
180:
176:
158:
154:
150:
147:
144:
139:
135:
114:
111:
108:
105:
102:
97:
93:
87:
83:
79:
76:
73:
68:
64:
58:
54:
45:
41:
37:
28:
22:
3992:Discriminant
3911:Multivariate
3831:
3751:
3732:
3712:
3692:
3669:
3662:
3649:
3642:
3615:
3603:
3553:function of
3546:real numbers
3543:
3402:, yielding
3397:
3393:
3385:
3381:
3378:
3370:affine space
3363:-dimensional
3349:
3330:
3316:
3312:inconsistent
3311:
3309:
3302:
3226:
3219:
3212:
3208:
3203:
3199:
3195:
3187:
3180:
3073:
3053:
3042:
2780:
2521:
2514:
2506:
2358:
2349:
2342:
2339:
2198:
1997:
1990:
1987:
1977:
1970:
1959:
1952:
1948:
1934:
1925:
1922:
1850:
1841:
1838:
1830:
1763:
1696:
1582:
1446:
1389:
1335:
1272:
1265:
1212:
1196:
1185:
1177:
1127:
1116:
1113:
1061:
1055:
1040:
1037:
1032:
1030:
1007:
953:
949:
945:
939:
935:
922:
918:
899:
893:
886:
880:
863:
777:
704:
650:
647:
625:
555:
511:
440:
437:One variable
419:real numbers
416:
401:
386:
373:
359:
354:
350:
315:
308:
297:
239:real numbers
235:coefficients
39:
33:
3941:Homogeneous
3936:Square-free
3931:Irreducible
3796:Polynomials
3637:, pp. 52-53
3550:real-valued
3204:coefficient
2517:determinant
1270:-intercept
1194:-intercept
901:linear maps
408:engineering
247:expressions
36:mathematics
4012:Categories
3901:Univariate
3685:References
3356:hyperplane
3045:hyperplane
1190:, and its
1125:, and, if
828:-intercept
382:hyperplane
302:over some
243:parameters
4023:Equations
3987:Resultant
3926:Trinomial
3906:Bivariate
3758:EMS Press
3487:≠
3469:…
3457:∈
3450:∑
3446:−
3426:−
3340:variables
3131:⋯
2882:−
2837:−
2802:−
2742:−
2717:−
2690:−
2672:−
2619:−
2597:−
2581:−
2565:−
2543:−
2463:−
2418:−
2383:−
2303:−
2281:−
2265:−
2249:−
2227:−
2168:−
2146:−
2121:−
2092:−
2050:−
2025:−
1803:−
1778:−
1733:−
1666:−
1546:−
1509:−
1472:−
1359:−
1305:−
1146:−
1082:−
837:−
801:−
750:−
734:−
526:−
494:≠
391:) in the
334:≠
311:solutions
270:…
208:…
175:variables
148:…
77:…
3972:Division
3921:Binomial
3916:Monomial
3610:, pg. 15
3563:See also
3342:are the
868:calculus
780:function
233:are the
179:unknowns
173:are the
44:equation
3760:, 2001
3622:, p. 25
3326:-tuples
423:complex
404:physics
355:unknown
181:), and
3809:degree
3721:
3699:
3358:in an
3222:> 0
782:. The
611:where
374:linear
127:where
42:is an
3595:Notes
3374:plane
3314:(for
3217:with
1203:(the
792:slope
790:with
784:graph
615:and
483:with
427:field
304:field
3798:and
3719:ISBN
3697:ISBN
3368:(or
3352:− 1)
3331:The
3280:and
1967:and
1932:and
1848:and
1697:or
1024:and
1018:line
874:, a
823:and
788:line
630:and
560:and
406:and
370:line
309:The
177:(or
38:, a
3807:By
3389:≠ 0
3319:≠ 0
3056:– 1
2199:By
1988:If
1130:= 0
1119:≠ 0
1058:= 0
1049:of
1043:≠ 0
1038:If
896:= 0
883:= 0
653:≠ 0
648:If
626:If
509:.
389:− 1
34:In
4014::
3756:,
3750:,
3627:^
3559:.
3376:.
3224:.
3166:0.
3066:.
3028:0.
2766:0.
2348:=
1996:≠
1976:,
1958:,
1908:1.
1111:.
952:,
938:=
921:=
904:.
623:.
548:.
433:.
399:.
357:.
3788:e
3781:t
3774:v
3555:n
3529:.
3524:i
3520:x
3512:j
3508:a
3502:i
3498:a
3490:j
3484:i
3481:,
3478:}
3475:n
3472:,
3466:,
3463:1
3460:{
3454:i
3439:j
3435:a
3431:b
3423:=
3418:j
3414:x
3398:j
3394:x
3386:j
3382:a
3361:n
3350:n
3348:(
3338:n
3333:n
3324:n
3317:b
3305:n
3288:z
3268:y
3264:,
3261:x
3241:3
3238:=
3235:n
3220:i
3213:i
3209:a
3191:0
3188:a
3183:b
3163:=
3160:b
3157:+
3152:n
3148:x
3142:n
3138:a
3134:+
3128:+
3123:2
3119:x
3113:2
3109:a
3105:+
3100:1
3096:x
3090:1
3086:a
3054:n
3049:n
3025:=
3020:|
3014:1
3007:2
3003:y
2995:2
2991:x
2983:1
2976:1
2972:y
2964:1
2960:x
2952:1
2947:y
2942:x
2936:|
2911:0
2908:=
2905:)
2900:1
2896:y
2890:2
2886:x
2877:2
2873:y
2867:1
2863:x
2859:(
2856:+
2853:y
2850:)
2845:1
2841:x
2832:2
2828:x
2824:(
2821:+
2818:x
2815:)
2810:2
2806:y
2797:1
2793:y
2789:(
2763:=
2758:|
2750:1
2746:y
2737:2
2733:y
2725:1
2721:x
2712:2
2708:x
2698:1
2694:y
2687:y
2680:1
2676:x
2669:x
2663:|
2638:0
2635:=
2632:)
2627:1
2623:x
2616:x
2613:(
2610:)
2605:1
2601:y
2592:2
2588:y
2584:(
2578:)
2573:1
2569:y
2562:y
2559:(
2556:)
2551:1
2547:x
2538:2
2534:x
2530:(
2492:0
2489:=
2486:)
2481:1
2477:y
2471:2
2467:x
2458:2
2454:y
2448:1
2444:x
2440:(
2437:+
2434:y
2431:)
2426:1
2422:x
2413:2
2409:x
2405:(
2402:+
2399:x
2396:)
2391:2
2387:y
2378:1
2374:y
2370:(
2353:2
2350:x
2346:1
2343:x
2325:,
2322:0
2319:=
2316:)
2311:1
2307:x
2300:x
2297:(
2294:)
2289:1
2285:y
2276:2
2272:y
2268:(
2262:)
2257:1
2253:y
2246:y
2243:(
2240:)
2235:1
2231:x
2222:2
2218:x
2214:(
2184:.
2181:)
2176:1
2172:x
2165:x
2162:(
2154:1
2150:x
2141:2
2137:x
2129:1
2125:y
2116:2
2112:y
2105:=
2100:1
2096:y
2089:y
2066:.
2058:1
2054:x
2045:2
2041:x
2033:1
2029:y
2020:2
2016:y
2001:2
1998:x
1994:1
1991:x
1983:)
1981:2
1978:y
1974:2
1971:x
1969:(
1965:)
1963:1
1960:y
1956:1
1953:x
1951:(
1938:0
1935:y
1929:0
1926:x
1905:=
1898:0
1894:y
1890:y
1885:+
1878:0
1874:x
1870:x
1854:0
1851:y
1845:0
1842:x
1816:)
1811:1
1807:x
1800:x
1797:(
1794:m
1791:=
1786:1
1782:y
1775:y
1749:.
1744:1
1740:x
1736:m
1728:1
1724:y
1720:+
1717:x
1714:m
1711:=
1708:y
1682:,
1679:)
1674:1
1670:x
1663:x
1660:(
1657:m
1654:+
1649:1
1645:y
1641:=
1638:y
1613:1
1609:y
1605:,
1600:1
1596:x
1585:m
1559:.
1554:b
1551:c
1543:=
1534:0
1530:y
1522:,
1517:a
1514:c
1506:=
1497:0
1493:x
1485:,
1480:b
1477:a
1469:=
1462:m
1432:,
1429:0
1426:=
1423:c
1420:+
1417:y
1414:b
1411:+
1408:x
1405:a
1375:.
1370:0
1366:x
1362:m
1356:x
1353:m
1350:=
1347:y
1321:,
1318:)
1313:0
1309:x
1302:x
1299:(
1296:m
1293:=
1290:y
1276:0
1273:x
1268:x
1251:.
1246:0
1242:y
1238:+
1235:x
1232:m
1229:=
1226:y
1209:y
1205:y
1200:0
1197:y
1192:y
1188:m
1159:.
1154:b
1151:c
1143:=
1140:y
1128:a
1123:y
1117:a
1109:x
1095:,
1090:a
1087:c
1079:=
1076:x
1066:y
1056:b
1051:x
1041:b
1026:b
1022:a
993:0
990:=
987:c
984:+
981:y
978:b
975:+
972:x
969:a
956:)
954:y
950:x
948:(
940:b
936:y
923:a
919:x
894:c
881:c
850:.
845:b
842:c
826:y
809:b
806:a
763:.
758:b
755:c
747:x
742:b
739:a
731:=
728:y
715:y
711:x
707:y
690:0
687:=
684:c
681:+
678:y
675:b
672:+
669:x
666:a
651:b
632:b
628:a
621:0
617:b
613:a
599:,
596:0
593:=
590:c
587:+
584:y
581:b
578:+
575:x
572:a
562:y
558:x
534:a
531:b
523:=
520:x
497:0
491:a
471:,
468:0
465:=
462:b
459:+
456:x
453:a
443:x
397:n
387:n
378:n
337:0
329:1
325:a
281:n
277:a
273:,
267:,
262:1
258:a
219:n
215:a
211:,
205:,
200:1
196:a
192:,
189:b
159:n
155:x
151:,
145:,
140:1
136:x
115:,
112:0
109:=
106:b
103:+
98:n
94:x
88:n
84:a
80:+
74:+
69:1
65:x
59:1
55:a
23:.
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