To solve the system of equations, we need to determine the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy the following system:
[tex]\[
\begin{cases}
-3x + 6y = 6 \\
7x + 6y = 6
\end{cases}
\][/tex]
We'll solve this system step-by-step.
1. Write the system of equations:
[tex]\[
\begin{cases}
-3x + 6y = 6 \quad \text{(Equation 1)} \\
7x + 6y = 6 \quad \text{(Equation 2)}
\end{cases}
\][/tex]
2. Isolate one of the variables:
Let's isolate [tex]\(y\)[/tex] by adding the two equations together. Notice that the coefficients of [tex]\(y\)[/tex] in both equations are the same, so when we add them, the [tex]\(y\)[/tex]-terms will cancel out:
[tex]\[
(-3x + 6y) + (7x + 6y) = 6 + 6
\][/tex]
3. Combine like terms:
[tex]\[
-3x + 7x + 6y + 6y = 12
\][/tex]
Simplifies to:
[tex]\[
4x + 12y = 12
\][/tex]
4. Solve for one variable:
Since [tex]\(y\)[/tex]-terms cancel out, this equation simplifies our problem significantly. We need to isolate [tex]\(x\)[/tex]:
[tex]\[
4x + 12 = 12
\][/tex]
Subtract 12 from both sides:
[tex]\[
4x = 0
\][/tex]
Divide by 4:
[tex]\[
x = 0
\][/tex]
5. Substitute back to find [tex]\(y\)[/tex]:
Now that we have [tex]\(x = 0\)[/tex], we can substitute [tex]\(x\)[/tex] back into one of the original equations to find [tex]\(y\)[/tex]. Let's use Equation 1:
[tex]\[
-3(0) + 6y = 6
\][/tex]
Simplifies to:
[tex]\[
6y = 6
\][/tex]
Dividing both sides by 6:
[tex]\[
y = 1
\][/tex]
6. State the solution:
The solution to the system of equations is:
[tex]\[
(x, y) = (0, 1)
\][/tex]
So, the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations are [tex]\(x = 0\)[/tex] and [tex]\(y = 1\)[/tex].
