Answer
Coincide.
Work Step by Step
To solve this system of equations, we use the graphing method.
$4x-6y=10$
$2x-3y=5$
Taking the first equation, we solve for y.
$4x-6y=10$
$-6y=10-4x$
$y=\frac{10-4x}{-6}$
Find three solutions:
For x=2,
$y=\frac{10-4(2)}{-6}$
$y=\frac{10-8}{-6}$
$y=\frac{1}{-3}\approx-0.33$
For x=0,
$y=\frac{10-4(0)}{-6}$
$y=\frac{10-0}{-6}$
$y=\frac{10}{-6}\approx-1.67$
For x=-2,
$y=\frac{10-4(-2)}{-6}$
$y=\frac{10+8}{-6}$
$y=\frac{18}{-6}=-3$
With the three points, $(2,-0.33), (0,-1.67), (-2,-3)$, we can graph the straight line that goes through these points.
Taking the second equation, we solve for y.
$2x-3y=5$
$-3y=5-2x$
$y=\frac{5-2x}{-3}$
Find three solutions:
For x=2,
$y=\frac{5-2(2)}{-3}$
$y=\frac{5-4}{-3}$
$y=\frac{1}{-3}\approx-0.33$
For x=0,
$y=\frac{5-2(0)}{-3}$
$y=\frac{5-0}{-3}$
$y=\frac{5}{-3}\approx-1.67$
For x=-2,
$y=\frac{5-2(-2)}{-3}$
$y=\frac{5+4}{-3}$
$y=\frac{9}{-3}\approx-3$
With the three points, $(2,-0.33), (0,-1.67), (-2,-3)$, we can graph the straight line that goes through these points.
The intersection point between these two lines is the answer to the system of equations.
In this case, the two lines go through the exact same points, so they coincide. This means that the system of equations has infinite solutions.
