Question

1. solve the system of equations by graphing.

7
$y = -x + 7$
$5x + 5y = 10$

1. solve the system of equations by graphing.

9
$y = \frac{3}{2}x - 2$
$y = 2x - 1$

1. draw a second line on the graph in order to demonstrate a system of equations with no solution. record your equation.

1. draw a second line on the graph in order to demonstrate a system of equations with one solution. record your equation.

Explanation:

Step1: Analyze Equation 1 ($y = -x + 7$)

For $y=-x + 7$, when $x = 0$, $y=7$ (y - intercept). When $y = 0$, $0=-x + 7\Rightarrow x = 7$ (x - intercept). Plot points $(0,7)$ and $(7,0)$.

Step2: Analyze Equation 2 ($5x + 5y = 10$)

Simplify $5x + 5y = 10$ by dividing by 5: $x + y = 2\Rightarrow y=-x + 2$. For $y=-x + 2$, when $x = 0$, $y = 2$; when $y = 0$, $x = 2$. Plot points $(0,2)$ and $(2,0)$.

Step3: Find Intersection

Graph both lines. The first line has slope $-1$, y - intercept $7$; the second has slope $-1$, y - intercept $2$. Wait, parallel lines? No, wait, miscalculation. Wait, $5x+5y = 10$: $5y=-5x + 10\Rightarrow y=-x + 2$. So both lines $y=-x + 7$ and $y=-x + 2$ have the same slope ($-1$), different y - intercepts. So they are parallel, no solution? Wait, but the problem is to solve by graphing. Wait, maybe I misread. Wait, the system is $y=-x + 7$ and $5x + 5y = 10$. Let's check intersection algebraically: substitute $y=-x + 7$ into $5x + 5y = 10$: $5x+5(-x + 7)=10\Rightarrow 5x-5x + 35 = 10\Rightarrow 35 = 10$, which is false. So no solution (parallel lines).

For problem 9: $y=\frac{3}{2}x - 2$ and $y = 2x - 1$.

Step1: Analyze $y=\frac{3}{2}x - 2$

Slope $\frac{3}{2}$, y - intercept $-2$. Plot $(0,-2)$, then from there up 3, right 2: $(2,1)$.

Step2: Analyze $y = 2x - 1$

Slope $2$, y - intercept $-1$. Plot $(0,-1)$, then up 2, right 1: $(1,1)$.

Step3: Find Intersection

Find x and y where $\frac{3}{2}x - 2=2x - 1$. Subtract $\frac{3}{2}x$: $-2=\frac{1}{2}x - 1$. Add 1: $-1=\frac{1}{2}x\Rightarrow x=-2$. Then $y=2(-2)-1=-5$. So intersection at $(-2,-5)$.

Answer:

For problem 7 (system $y=-x + 7$ and $5x + 5y = 10$): No solution (parallel lines).
For problem 9 (system $y=\frac{3}{2}x - 2$ and $y = 2x - 1$): Solution is $x=-2$, $y=-5$ (or the point $(-2,-5)$).

(Note: Since the user's image has multiple problems, here we solved problem 7 and 9 as examples of solving systems by graphing. The key steps involve finding intercepts or using slope - intercept form to plot lines and find their intersection (or determine parallelism for no solution).)