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:
[tex]\begin{gathered} y=\frac{3}{2}x-4 \\ y=\frac{3}{2}\cdot0-4 \\ y=0-4 \\ y=-4 \end{gathered}[/tex]Then, the line passes through (0, -4)
Substituting with x = 2 into y = 3/2x - 4, we get:
[tex]\begin{gathered} y=\frac{3}{2}\cdot2-4 \\ y=3-4 \\ y=-1 \end{gathered}[/tex]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:
[tex]\begin{gathered} y=-\frac{1}{2}x+4 \\ y=\frac{3}{2}x-4 \end{gathered}[/tex]At the solution, both y-values are equal, then:
[tex]\begin{gathered} -\frac{1}{2}x+4=\frac{3}{2}x-4 \\ -\frac{1}{2}x+4+\frac{1}{2}x=\frac{3}{2}x-4+\frac{1}{2}x \\ 4=2x-4 \\ 4+4=2x-4+4 \\ 8=2x \\ \frac{8}{2}=\frac{2x}{2} \\ 4=x \end{gathered}[/tex]Substituting x = 4 into the first equation, we get:
[tex]\begin{gathered} y=-\frac{1}{2}\cdot4+4 \\ y=-2+4 \\ y=2 \end{gathered}[/tex]This solution coincides with the first one.
