In the xy-plane, the graphs of the linear equations y = 3x - 5 and y = -x + 7 intersect at… — SAT

In the $xy$ -plane, the graphs of the linear equations $y = 3x - 5$ and $y = -x + 7$ intersect at the point $(x, y)$ . What is the value of $x$ ?

To find the

x

-value of the intersection point of the two graphs, set the two equations equal to each other because both represent the same

y

-value at that point:

3x - 5 = -x + 7

. Adding

x

to both sides of the equation gives

4x - 5 = 7

. Adding

5

to both sides gives

4x = 12

. Dividing both sides by

4

yields

x = 3

Set the two expressions for

y

equal to each other to find the

x

-coordinate of the intersection point.

3x - 5 = -x + 7

Since both equations are solved for

y

, their right-hand sides must be equal at the point of intersection.

Add

x

to both sides of the equation to collect the variable terms on one side.

4x - 5 = 7

Moving the variable terms together allows us to isolate

x

Add

5

to both sides of the equation to isolate the term with

x

4x = 12

Grouping the constant terms on the opposite side prepares the equation for final division.

Divide both sides by

4

to find the value of

x

x = 3

This isolates

x

and gives the final solution.

Solving a system of linear equations by setting the equations equal to find the point of intersection in the coordinate plane.

Question