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Sagot :
To solve the system of equations using the elimination method, we'll follow these steps:
1. Write down the given system of equations:
[tex]\[ \begin{cases} 5x + 8y = -88 \\ x - y = -2 \end{cases} \][/tex]
2. Eliminate one of the variables:
To eliminate the variable [tex]\(x\)[/tex], we can first ensure that the coefficient of [tex]\(x\)[/tex] in the second equation is the same as in the first. We'll make the coefficient of [tex]\(x\)[/tex] in the second equation equal to 5 by multiplying the entire second equation by 5:
[tex]\[ 5(x - y) = 5(-2) \][/tex]
This gives us the new system:
[tex]\[ \begin{cases} 5x + 8y = -88 \\ 5x - 5y = -10 \end{cases} \][/tex]
3. Subtract the second modified equation from the first to eliminate [tex]\(x\)[/tex]:
[tex]\[ \begin{aligned} (5x + 8y) - (5x - 5y) &= -88 - (-10) \\ 5x + 8y - 5x + 5y &= -88 + 10 \\ 13y &= -78 \end{aligned} \][/tex]
Simplifying this, we find:
[tex]\[ y = \frac{-78}{13} = -6 \][/tex]
4. Substitute [tex]\(y\)[/tex] back into one of the original equations to solve for [tex]\(x\)[/tex]:
We'll use the second equation for substitution:
[tex]\[ x - y = -2 \][/tex]
Substitute [tex]\(y = -6\)[/tex]:
[tex]\[ x - (-6) = -2 \\ x + 6 = -2 \\ x = -2 - 6 \\ x = -8 \][/tex]
5. Write the solution as an ordered pair:
[tex]\[ (x, y) = (-8, -6) \][/tex]
Thus, the solution to the system of equations is:
[tex]\[ (-8, -6) \][/tex]
1. Write down the given system of equations:
[tex]\[ \begin{cases} 5x + 8y = -88 \\ x - y = -2 \end{cases} \][/tex]
2. Eliminate one of the variables:
To eliminate the variable [tex]\(x\)[/tex], we can first ensure that the coefficient of [tex]\(x\)[/tex] in the second equation is the same as in the first. We'll make the coefficient of [tex]\(x\)[/tex] in the second equation equal to 5 by multiplying the entire second equation by 5:
[tex]\[ 5(x - y) = 5(-2) \][/tex]
This gives us the new system:
[tex]\[ \begin{cases} 5x + 8y = -88 \\ 5x - 5y = -10 \end{cases} \][/tex]
3. Subtract the second modified equation from the first to eliminate [tex]\(x\)[/tex]:
[tex]\[ \begin{aligned} (5x + 8y) - (5x - 5y) &= -88 - (-10) \\ 5x + 8y - 5x + 5y &= -88 + 10 \\ 13y &= -78 \end{aligned} \][/tex]
Simplifying this, we find:
[tex]\[ y = \frac{-78}{13} = -6 \][/tex]
4. Substitute [tex]\(y\)[/tex] back into one of the original equations to solve for [tex]\(x\)[/tex]:
We'll use the second equation for substitution:
[tex]\[ x - y = -2 \][/tex]
Substitute [tex]\(y = -6\)[/tex]:
[tex]\[ x - (-6) = -2 \\ x + 6 = -2 \\ x = -2 - 6 \\ x = -8 \][/tex]
5. Write the solution as an ordered pair:
[tex]\[ (x, y) = (-8, -6) \][/tex]
Thus, the solution to the system of equations is:
[tex]\[ (-8, -6) \][/tex]
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