To solve the system of equations:
[tex]\[
\begin{cases}
2x + 6y = 6 \\
3x - 2y = 20
\end{cases}
\][/tex]
we will use the method of elimination or substitution. Let's start by simplifying and solving this step-by-step:
1. Rewrite the equations for clarity:
[tex]\[
2x + 6y = 6 \quad \text{(Equation 1)}
\][/tex]
[tex]\[
3x - 2y = 20 \quad \text{(Equation 2)}
\][/tex]
2. Eliminate one of the variables.
We can multiply Equation 2 by 3 to make the coefficients of [tex]\( y \)[/tex] equal but opposite in sign:
[tex]\[
3(3x - 2y) = 3(20)
\][/tex]
[tex]\[
9x - 6y = 60 \quad \text{(Equation 3)}
\][/tex]
3. Add Equation 1 and Equation 3 to eliminate [tex]\( y \)[/tex]:
[tex]\[
(2x + 6y) + (9x - 6y) = 6 + 60
\][/tex]
[tex]\[
2x + 9x + 6y - 6y = 66
\][/tex]
[tex]\[
11x = 66
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{66}{11}
\][/tex]
[tex]\[
x = 6
\][/tex]
5. Substitute [tex]\( x = 6 \)[/tex] back into one of the original equations to solve for [tex]\( y \)[/tex]:
Using Equation 1:
[tex]\[
2(6) + 6y = 6
\][/tex]
[tex]\[
12 + 6y = 6
\][/tex]
[tex]\[
6y = 6 - 12
\][/tex]
[tex]\[
6y = -6
\][/tex]
[tex]\[
y = \frac{-6}{6}
\][/tex]
[tex]\[
y = -1
\][/tex]
6. Write the solution as an ordered pair:
The solution to the system of equations is:
[tex]\[
(x, y) = (6, -1)
\][/tex]
7. Verify the solution by substituting [tex]\( x = 6 \)[/tex] and [tex]\( y = -1 \)[/tex] back into the original equations:
[tex]\[
2(6) + 6(-1) = 12 - 6 = 6 \quad \text{(True for Equation 1)}
\][/tex]
[tex]\[
3(6) - 2(-1) = 18 + 2 = 20 \quad \text{(True for Equation 2)}
\][/tex]
Since both equations are satisfied, the solution is correct. Now, look at the given choices.
- A. [tex]\((6, -1)\)[/tex]
- B. [tex]\((3, 0)\)[/tex]
- C. [tex]\((6, -2)\)[/tex]
- D. [tex]\((3, 2)\)[/tex]
The correct choice is A. [tex]\((6, -1)\)[/tex].
