Question

The linear equation y=-1/2x+4 is shown graphed. If it was combined with the equation y=3/2x-4 to form a system, what would be the solution to this system? Show how you arrived at your answer

Answer 1

To solve the system of equations, we need to graph y = 3/2x - 4 into the same cartesian plane of the graph of y = -1/2x + 4.

We need two points to graph a line. Substituting with x = 0 into y = 3/2x - 4, we get:

`\begin{gathered} y=(3)/(2)x-4 \\ y=(3)/(2)\cdot0-4 \\ y=0-4 \\ y=-4 \end{gathered}`

Then, the line passes through (0, -4)

Substituting with x = 2 into y = 3/2x - 4, we get:

`\begin{gathered} y=(3)/(2)\cdot2-4 \\ y=3-4 \\ y=-1 \end{gathered}`

Then, the line passes through (2, -1).

Connecting these two points, y = 3/2x - 4 is graphed.

y = -1/2x + 4 is in red and y = 3/2x - 4 is in blue.

The point at which both lines intersect is the solution to the system. From the graph, the solution is (4, 2) or x = 4 and y = 2

We can solve this system of equations algebraically. We have the next two equations:

`\begin{gathered} y=-(1)/(2)x+4 \\ y=(3)/(2)x-4 \end{gathered}`

At the solution, both y-values are equal, then:

`\begin{gathered} -(1)/(2)x+4=(3)/(2)x-4 \\ -(1)/(2)x+4+(1)/(2)x=(3)/(2)x-4+(1)/(2)x \\ 4=2x-4 \\ 4+4=2x-4+4 \\ 8=2x \\ (8)/(2)=(2x)/(2) \\ 4=x \end{gathered}`

Substituting x = 4 into the first equation, we get:

`\begin{gathered} y=-(1)/(2)\cdot4+4 \\ y=-2+4 \\ y=2 \end{gathered}`

This solution coincides with the first one.