Question

solve each system by graphing. when you finish, the remaining letters will tell you the answer to the title question.

1. $y = \frac{3}{2}x - 1$; $y = -x + 4$
2. $y = \frac{1}{3}x + 2$; $y = -\frac{4}{3}x - 3$
3. $y = 2x + 1$; $-2x + 3y = -9$
4. $3x + y = 0$; $x - y = 4$
5. $-3x + 4y = 8$; $x + 2y + 6 = 0$
6. $7x - 5y = 20$; $-8x - 3y = 12$
7. $-x - 4y = 12$; $20x + 80y = 0$
8. $30x + 50y - 100 = 0$; $3x - 15y - 30 = 0$

Explanation:

Step1: Analyze the first equation

The first equation is \( y = \frac{3}{2}x - 1 \). To find the y - intercept, set \( x = 0 \), then \( y=-1 \). To find the x - intercept, set \( y = 0 \), then \( 0=\frac{3}{2}x - 1\), \(\frac{3}{2}x=1\), \(x=\frac{2}{3}\approx0.67\).

Step2: Analyze the second equation

The second equation is \( y=-x + 4 \). The y - intercept is \( y = 4 \) (when \( x = 0 \)), and the x - intercept is \( x = 4 \) (when \( y = 0 \)).

Step3: Graph the lines

For \( y=\frac{3}{2}x - 1 \), start at the y - intercept \((0,-1)\) and use the slope \(\frac{3}{2}\) (rise 3, run 2) to plot other points. For \( y=-x + 4 \), start at the y - intercept \((0,4)\) and use the slope \(- 1\) (rise - 1, run 1) to plot other points. The intersection point of the two lines is the solution of the system. By graphing (or solving the system \(

$$\begin{cases}y=\frac{3}{2}x - 1\\y=-x + 4\end{cases}$$

\), set \(\frac{3}{2}x-1=-x + 4\), \(\frac{3}{2}x+x=4 + 1\), \(\frac{3x + 2x}{2}=5\), \(\frac{5x}{2}=5\), \(x = 2\), then \(y=-2 + 4=2\)), the solution is \((2,2)\). Looking at the table on the left, the letter corresponding to \((2,2)\) is "OP".

Answer:

OP