To determine the solution to the given system of equations, we need to find the point at which the two lines intersect. Here are the equations:
1. [tex]\( y = 3x + 9 \)[/tex]
2. [tex]\( 6x + 2y = 6 \)[/tex]
### Step-by-Step Solution
#### Step 1: Convert the Second Equation to Slope-Intercept Form
The second equation is [tex]\( 6x + 2y = 6 \)[/tex]. To express this equation in the slope-intercept form [tex]\( y = mx + b \)[/tex], follow these steps:
1. Subtract [tex]\( 6x \)[/tex] from both sides:
[tex]\[
2y = 6 - 6x
\][/tex]
2. Divide every term by 2:
[tex]\[
y = -3x + 3
\][/tex]
Now, we have the two equations in slope-intercept form:
- [tex]\( y = 3x + 9 \)[/tex]
- [tex]\( y = -3x + 3 \)[/tex]
#### Step 2: Find the Intersection Point
To find the point of intersection, set the right-hand sides of the equations equal to each other:
[tex]\[
3x + 9 = -3x + 3
\][/tex]
Solve for [tex]\( x \)[/tex]:
1. Add [tex]\( 3x \)[/tex] to both sides:
[tex]\[
6x + 9 = 3
\][/tex]
2. Subtract 9 from both sides:
[tex]\[
6x = -6
\][/tex]
3. Divide by 6:
[tex]\[
x = -1
\][/tex]
With [tex]\( x = -1 \)[/tex], substitute this value into one of the original equations to find [tex]\( y \)[/tex]. Using [tex]\( y = 3x + 9 \)[/tex]:
[tex]\[
y = 3(-1) + 9 = -3 + 9 = 6
\][/tex]
Therefore, the solution to the system is [tex]\( (-1, 6) \)[/tex].
### Conclusion
The system of equations has one unique solution: [tex]\( (-1, 6) \)[/tex].
The graphical representations of these equations will intersect at this point. Consequently, the correct answer is:
There is one unique solution [tex]\((-1, 6)\)[/tex].
