...Ant. 4. 38. To find the condition that three right lines snail meet in a point. Let their equations be Ax + By + C= 0, Ax + By + C' = 0, A"x + B"y + C" = 0. If they intersect, the coordinates of the intersection of two of them must satisfy the third equation....

...equation of condition between the nine parameters of the three lines. Let the equations of the lines be ax + by +c = 0; a'x + b'y +c' =0; a"x + b''y + c" = 0. If the three lines intersect in a point, there must be one pair of values of x and y which satisfy...

...4y-25 = 0. This is the equation of the required line 53. If the equations of three straight lines are Ax +By +C = 0, A'x + B'y +C" = 0, A"x + B"y + C" =0, and we can find three constants, I, m, n, so that the relation l(Ax+£y+ C) + m (A'x +B'y + C") + n...

...0 (1) will represent a straight line parallel to them; for (L), (M), (N), &c. will be of the forms Ax + By + C= 0, Ax + By + C' = 0, Ax + By + C" = 0, &c. and equation (1) becomes Ax + By + lC^C^nC^ l + m + n + &c. which represents a straight line parallel...

..................... (1) will represent a straight line parallel to them; for (L), (M), (N), &c. will be of the forms Ax + By + C= 0, Ax + By + C' = 0, Ax + By + C" = 0, &c. and equation (1) becomes , , j, , lC+mC' + nC" + &c. . Ax + By l -- j- - —p - = 0, í + m + n...

...satisfies all the equations of the system or, as it is commonly put, a system of more than two cquatiom in two variables x and y is in general inconsistent....the particular case of a linear system, say — ax + Ъу + с = 0, a'x + Ь'у + с = 0, (37), a"'x + Ъ'"у + с" = О, etc. a definite proof may be...

...+ b'y + c' = 0 is transformed into the axis y, = 0. Hence, the triangle formed by the three lines, ax + by + c = 0, a'x + b'y + c' = 0, a"x + b"y + c" = 0, is transformed into the triangle formed by the coordinate axes and the line at infinity. That the first...

...and (/3) is therefore 11 2 * = I6' * = 6' § 8.] Three equations of the 1st degree in two variables, say • ax + by + c = 0, a'x + b'y + c' = 0, a"x + b"y + c" = 0 ( 1 ), will not be consistent unless a"(bc' - b'c) + V'(ai - c'a) + c"(ab' - a'b) = 0 (2) ; and tJiey...

...in a plane have not in general a common point of intersection, it follows that the three equations, ax + by + c = 0, a'x + b'y + c' = 0, a"x + b"y + c" = 0 (1 ), have not in general a common solution. When these have a common solution their three graphic...

...triangle rapporté à un système d'axes rectangulaires se rencontrant en son intérieur, et soient ax + by + c = 0, a'x + b'y + c' = 0, a"x + b"y + c" = 0 les équations des côtés BC, CA, AB. Lorsque les côtés du triangle sont associés à des directions...