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Question 4 of 10

Solve the following system of equations. Express your answer as an ordered pair in the format [tex]$(a, b)$[/tex], with no spaces between the numbers or symbols.

[tex]\[
\begin{array}{l}
2x + 7y = -7 \\
-4x - 3y = -19
\end{array}
\][/tex]

Answer here:

Sagot :

GinnyAnswer
To solve the given system of equations:

[tex]\[ \begin{cases} 2x + 7y = -7 \\ -4x - 3y = -19 \end{cases} \][/tex]

we need to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that simultaneously satisfy both equations. Here’s a detailed step-by-step process:

1. Label the equations:
[tex]\[ \begin{aligned} &\text{(1)} \quad 2x + 7y = -7 \\ &\text{(2)} \quad -4x - 3y = -19 \end{aligned} \][/tex]

2. Multiply the equations to align coefficients for elimination:
We'll multiply the first equation by 2 to match the coefficients of [tex]\( x \)[/tex] in magnitude with that in the second equation:
[tex]\[ 2 \times (2x + 7y) = 2 \times (-7) \][/tex]
This gives us:
[tex]\[ 4x + 14y = -14 \quad \text{(3)} \][/tex]

3. Add equation (3) to equation (2):
[tex]\[ (4x + 14y) + (-4x - 3y) = -14 + (-19) \][/tex]
Simplify the left and right sides:
[tex]\[ (4x - 4x) + (14y - 3y) = -14 - 19 \][/tex]
This simplifies to:
[tex]\[ 11y = -33 \][/tex]

4. Solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{-33}{11} = -3 \][/tex]

5. Substitute [tex]\( y = -3 \)[/tex] back into equation (1) to find [tex]\( x \)[/tex]:
[tex]\[ 2x + 7(-3) = -7 \][/tex]
Simplify:
[tex]\[ 2x - 21 = -7 \][/tex]
Add 21 to both sides:
[tex]\[ 2x = 14 \][/tex]
Divide by 2:
[tex]\[ x = 7 \][/tex]

The solution to the system of equations is [tex]\( x = 7 \)[/tex] and [tex]\( y = -3 \)[/tex]. Expressed as an ordered pair, the solution is:

[tex]\[ (7,-3) \][/tex]
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