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Select the correct answer.

Solve the system of equations below.

[tex]\[
\begin{aligned}
-3x + 6y &= 9 \\
5x + 7y &= -49
\end{aligned}
\][/tex]

A. [tex]\((-7, -2)\)[/tex]
B. [tex]\((1, -2)\)[/tex]
C. [tex]\(\left(-2, \frac{1}{2}\right)\)[/tex]
D. [tex]\((-2, -7)\)[/tex]


Sagot :

To solve the system of equations

[tex]\[ \begin{aligned} -3x + 6y & = 9 \\ 5x + 7y & = -49 \end{aligned} \][/tex]

we need to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy both equations simultaneously.

Step 1: Simplify the first equation
[tex]\[ -3x + 6y = 9 \][/tex]

Divide every term by 3:
[tex]\[ -x + 2y = 3 \][/tex]

This is the simplified form of the first equation:
[tex]\[ x - 2y = -3 \quad \text{(Equation 1)} \][/tex]

Step 2: Multiply the first equation to make the coefficients of [tex]\( x \)[/tex] equal in both equations. We multiply Equation 1 by 5:
[tex]\[ 5(x - 2y) = 5(-3) \][/tex]
[tex]\[ 5x - 10y = -15 \quad \text{(Equation 3)} \][/tex]

Step 3: Write down the second original equation:
[tex]\[ 5x + 7y = -49 \quad \text{(Equation 2)} \][/tex]

Step 4: Subtract Equation 2 from Equation 3 to eliminate [tex]\( x \)[/tex]:
[tex]\[ (5x - 10y) - (5x + 7y) = -15 - (-49) \][/tex]
[tex]\[ 5x - 10y - 5x - 7y = 34 \][/tex]
[tex]\[ -17y = 34 \][/tex]

Step 5: Solve for [tex]\( y \)[/tex]:
[tex]\[ y = -2 \][/tex]

Step 6: Substitute [tex]\( y = -2 \)[/tex] back into Equation 1:
[tex]\[ x - 2(-2) = -3 \][/tex]
[tex]\[ x + 4 = -3 \][/tex]
[tex]\[ x = -7 \][/tex]

Therefore, the solution to the system of equations is:
[tex]\[ (x, y) = (-7, -2) \][/tex]

The correct answer is [tex]\(\boxed{(-7, -2)}\)[/tex].