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Solve the system of equations and choose the correct ordered pair.

[tex]
\begin{array}{l}
2x + 6y = 6 \\
3x - 2y = 20
\end{array}
[/tex]

A. [tex]$(6, -1)$[/tex]
B. [tex]$(3, 0)$[/tex]
C. [tex]$(6, -2)$[/tex]
D. [tex]$(3, 2)$[/tex]

Sagot :

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].