To solve this system of equations, we need to find the point at which the two equations intersect. Here is the step-by-step procedure:
1. Start with the given equations:
[tex]\[
\begin{array}{l}
y = 3x + 15 \\
3x + 3y = 9
\end{array}
\][/tex]
2. Convert the second equation into the slope-intercept form [tex]\( y = mx + b \)[/tex]:
Begin with the equation:
[tex]\[
3x + 3y = 9
\][/tex]
To isolate [tex]\( y \)[/tex], first subtract [tex]\( 3x \)[/tex] from both sides:
[tex]\[
3y = -3x + 9
\][/tex]
Next, divide each term by 3:
[tex]\[
y = -x + 3
\][/tex]
Now, we have the two equations:
[tex]\[
\begin{array}{l}
y = 3x + 15 \\
y = -x + 3
\end{array}
\][/tex]
3. Set the two equations equal to find the x-coordinate of the intersection point:
[tex]\[
3x + 15 = -x + 3
\][/tex]
Solve for [tex]\( x \)[/tex] by adding [tex]\( x \)[/tex] to both sides:
[tex]\[
3x + x + 15 = 3
\][/tex]
Simplify:
[tex]\[
4x + 15 = 3
\][/tex]
Subtract 15 from both sides:
[tex]\[
4x = 3 - 15
\][/tex]
Simplify:
[tex]\[
4x = -12
\][/tex]
Divide both sides by 4:
[tex]\[
x = -3
\][/tex]
4. Substitute [tex]\( x = -3 \)[/tex] back into one of the original equations to find the y-coordinate:
Using [tex]\( y = 3x + 15 \)[/tex]:
[tex]\[
y = 3(-3) + 15
\][/tex]
Simplify:
[tex]\[
y = -9 + 15
\][/tex]
Result:
[tex]\[
y = 6
\][/tex]
5. Summarize the intersection point:
The unique solution is the point where both lines intersect, which is:
[tex]\[
(-3, 6)
\][/tex]
Therefore, the solution to the system of equations is [tex]\(\boxed{(-3, 6)}\)[/tex]. There is one unique solution.
