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On your own paper, solve the system of equations using elimination and identify the solution. Always list your answers alphabetically in the ordered pairs.

[tex]\[
\begin{array}{l}
6x + 4y = 42 \\
-3x + 3y = -6
\end{array}
\][/tex]

A. (27, 30)
B. (3.6, 4.6)
C. (4.6, 3.6)
D. (3, 5)
E. (5, 3)
F. (30, 27)


Sagot :

Sure! Let's solve the given system of equations using the elimination method and find the correct solution.

We are given:
[tex]\[ \begin{array}{l} 6x + 4y = 42 \quad \text{... (1)} \\ -3x + 3y = -6 \quad \text{... (2)} \end{array} \][/tex]

Our goal is to eliminate one of the variables. In this case, let's eliminate [tex]\(x\)[/tex] by aligning the coefficients of [tex]\(x\)[/tex]. To do this, we can manipulate equation (2) so that the coefficient of [tex]\(x\)[/tex] matches the coefficient in equation (1).

Step 1: Multiply equation (2) by 2. This gives us:
[tex]\[ -3x \cdot 2 + 3y \cdot 2 = -6 \cdot 2 \][/tex]
which simplifies to:
[tex]\[ -6x + 6y = -12 \quad \text{... (3)} \][/tex]

Step 2: Now add equation (1) and equation (3):
[tex]\[ (6x + 4y) + (-6x + 6y) = 42 + (-12) \][/tex]
Simplifying this, we get:
[tex]\[ 6x - 6x + 4y + 6y = 42 - 12 \][/tex]
[tex]\[ 0x + 10y = 30 \][/tex]
[tex]\[ 10y = 30 \][/tex]

Step 3: Solve for [tex]\(y\)[/tex]:
[tex]\[ y = \frac{30}{10} \][/tex]
[tex]\[ y = 3 \][/tex]

Step 4: Substitute the value of [tex]\(y\)[/tex] back into equation (1) to solve for [tex]\(x\)[/tex]:
[tex]\[ 6x + 4y = 42 \][/tex]
Substitute [tex]\(y = 3\)[/tex]:
[tex]\[ 6x + 4 \cdot 3 = 42 \][/tex]
[tex]\[ 6x + 12 = 42 \][/tex]
Subtract 12 from both sides:
[tex]\[ 6x = 30 \][/tex]
Divide both sides by 6:
[tex]\[ x = \frac{30}{6} \][/tex]
[tex]\[ x = 5 \][/tex]

Therefore, the solution to the system of equations is:
[tex]\[ (x, y) = (5, 3) \][/tex]

Let's verify the result against the given multiple-choice answers:
- (27,30)
- (3.6,4.6)
- (4.6,3.6)
- (3,5)
- (5,3)
- (30,27)

The solution [tex]\((x, y) = (5, 3)\)[/tex] matches one of the options exactly:
[tex]\[ (5, 3) \][/tex]

Hence, the correct solution to the system of equations is:
[tex]\((5, 3)\)[/tex].