Graph the following system of equations:

[tex]\[
\begin{array}{l}
y = 3x + 9 \\
6x + 2y = 6
\end{array}
\][/tex]

What is the solution to the system?

A. There is no solution.
B. There is one unique solution [tex]\((-1, 6)\)[/tex].
C. There is one unique solution [tex]\((0, 3)\)[/tex].
D. There are infinitely many solutions.



Answer :

Let's analyze and solve the system of equations given:

[tex]\[ \begin{aligned} & \text{Equation 1: } y = 3x + 9 \\ & \text{Equation 2: } 6x + 2y = 6 \end{aligned} \][/tex]

### Step 1: Graph each equation

First, let's rewrite Equation 2 in slope-intercept form [tex]\(y = mx + b\)[/tex].

#### Equation 2:
[tex]\[ 6x + 2y = 6 \][/tex]

Subtract [tex]\(6x\)[/tex] from both sides:
[tex]\[ 2y = -6x + 6 \][/tex]

Divide by 2:
[tex]\[ y = -3x + 3 \][/tex]

Now we have both equations in the form [tex]\(y = mx + b\)[/tex]:

1. [tex]\(y = 3x + 9\)[/tex]
2. [tex]\(y = -3x + 3\)[/tex]

### Step 2: Graph the lines

Line 1: [tex]\(y = 3x + 9\)[/tex]
- The slope is 3, and the y-intercept is 9.
- This line rises steeply and crosses the y-axis at (0, 9).

Line 2: [tex]\(y = -3x + 3\)[/tex]
- The slope is -3, and the y-intercept is 3.
- This line falls steeply and crosses the y-axis at (0, 3).

### Step 3: Determine the intersection point

To find the intersection point, if any, we set the two equations equal to each other:

[tex]\[ 3x + 9 = -3x + 3 \][/tex]

Solve for [tex]\(x\)[/tex]:

Add [tex]\(3x\)[/tex] to both sides:
[tex]\[ 6x + 9 = 3 \][/tex]

Subtract 9 from both sides:
[tex]\[ 6x = -6 \][/tex]

Divide by 6:
[tex]\[ x = -1 \][/tex]

Now, substitute [tex]\(x = -1\)[/tex] back into either original equation to find [tex]\(y\)[/tex]. Using [tex]\(y = 3x + 9\)[/tex]:

[tex]\[ y = 3(-1) + 9 \][/tex]
[tex]\[ y = -3 + 9 \][/tex]
[tex]\[ y = 6 \][/tex]

Thus, the unique solution to this system is [tex]\((-1, 6)\)[/tex].

### Step 4: Verify the type of solution

Since both lines are straight and have different slopes, they intersect at exactly one point. There are no other intersection points.

### Conclusion

Given the graphs and calculations:
- There is one unique solution [tex]\((-1, 6)\)[/tex].

So, the solution to the system of equations is [tex]\(\boxed{(-1, 6)}\)[/tex].