Question

problems 1–2: here is the graph of this system of equations: \\(\

$$\begin{cases} y = -3x + 8 \\\\ y = 3x + 2 \\end{cases}$$

\\)

1. how can you determine the solution to this system of equations by looking at the graph?
2. what is the solution to the system of equations?

problems 3–5: use the lines on the graph to decide whether each statement is true or false.

1. the solution to the equation \\(-2x + 1 = 3x - 9\\) is \\(x = 2\\).
2. the point \\((2, -3)\\) is a solution to this system of equations: \\(\

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

\\)

1. the point \\((0, 1)\\) is a solution to the equation \\(y = -2x + 1\\).

Explanation:

Step1: Identify solution from graph

The solution of a system of linear equations graphed is the coordinates of the point where the two lines intersect.

Step2: Solve first system algebraically

Set $-3x + 8 = 3x + 2$
$-3x - 3x = 2 - 8$
$-6x = -6$
$x = 1$
Substitute $x=1$ into $y=3x+2$: $y=3(1)+2=5$

Step3: Verify equation solution

For $-2x + 1 = 3x -9$, solve for $x$:
$-2x -3x = -9 -1$
$-5x = -10$
$x=2$, match the statement.

Step4: Check point for system

For

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

, substitute $x=2$: $y=-2(2)+1=-3$, so $(2,-3)$ is a solution.

Step5: Verify point in equation

Substitute $(0,1)$ into $y=-2x+1$: $1=-2(0)+1=1$, which holds true.

Answer:

1. The solution is the coordinates of the intersection point of the two lines on the graph.
2. $(1, 5)$
3. True
4. True
5. True