How do you find where two lines meet on a graph by solving an equation?

Answer 1

You can't find this out by solving an equation, but you can by solving a system of two linear equations. Basically, you would need to know the equations for both of the lines before you began. Then, you would solve one of those linear equations for one of the variables in terms of the other, like y in terms of x. Then, you'd plug that value of y, in our case, into the other equation and solve for x. Then you'd plug that value of x, which is now a number, back into the original equation to get a numerical value for y. Your intersection point would then be the ordered pair (x,y). Here's an example:

2x + 14y = 8,

-6x + 10y = 4.

Solve the first equation for y:

y = (4/7) - (1/7)x.

Plug that value for y into the second equation:

-6x + (40/7) - (10/7)x = 4.

Solve for x:

(10/7)x + (42/7)x = (40/7) - (28/7),

(52/7)x = (12/7),

x = (12/52) = (6/26) = (3/13).

Plug that value of x into the first equation:

(6/13) + 14y = 8,

14y = (98/13),

y = (98/182) = (49/91) = (7/13).

So the point at which those two lines intersect would be the ordered pair (3/13,7/13).

Finally, it's usually worth while to verify your answer by plugging these values into the original equations.

2(3/13) + 14(7/13) = (6/13) + (98/13) = (104/13) = 8, so the first one's right.

-6(3/13) + 10(7/13) = -(18/13) + (70/13) = (52/13) = 4, so that one's right too.