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Use the substitution method to solve the system x=y-4,x+8y=2.Asked by T-Swizzle
Use the substitution method to solve the system: x = y - 4, x + 8y = 2
Answers
Answered by
GPT-4o mini
To solve the system of equations using the substitution method, we can start with the two given equations:
1. \( x = y - 4 \)
2. \( x + 8y = 2 \)
Since the first equation expresses \( x \) in terms of \( y \), we can substitute this expression into the second equation.
Substituting \( x = y - 4 \) into the second equation:
\[
(y - 4) + 8y = 2
\]
Now, we will simplify this equation:
\[
y - 4 + 8y = 2
\]
Combine the \( y \) terms:
\[
9y - 4 = 2
\]
Now add 4 to both sides:
\[
9y = 2 + 4
\]
\[
9y = 6
\]
Now, divide both sides by 9 to solve for \( y \):
\[
y = \frac{6}{9} = \frac{2}{3}
\]
Now that we have the value of \( y \), we can substitute it back into the first equation to find \( x \):
\[
x = y - 4
\]
\[
x = \frac{2}{3} - 4
\]
\[
x = \frac{2}{3} - \frac{12}{3} = \frac{2 - 12}{3} = \frac{-10}{3}
\]
Thus, the solution to the system of equations is:
\[
x = -\frac{10}{3}, \quad y = \frac{2}{3}
\]
The final answer is:
\[
(x, y) = \left(-\frac{10}{3}, \frac{2}{3}\right)
\]
1. \( x = y - 4 \)
2. \( x + 8y = 2 \)
Since the first equation expresses \( x \) in terms of \( y \), we can substitute this expression into the second equation.
Substituting \( x = y - 4 \) into the second equation:
\[
(y - 4) + 8y = 2
\]
Now, we will simplify this equation:
\[
y - 4 + 8y = 2
\]
Combine the \( y \) terms:
\[
9y - 4 = 2
\]
Now add 4 to both sides:
\[
9y = 2 + 4
\]
\[
9y = 6
\]
Now, divide both sides by 9 to solve for \( y \):
\[
y = \frac{6}{9} = \frac{2}{3}
\]
Now that we have the value of \( y \), we can substitute it back into the first equation to find \( x \):
\[
x = y - 4
\]
\[
x = \frac{2}{3} - 4
\]
\[
x = \frac{2}{3} - \frac{12}{3} = \frac{2 - 12}{3} = \frac{-10}{3}
\]
Thus, the solution to the system of equations is:
\[
x = -\frac{10}{3}, \quad y = \frac{2}{3}
\]
The final answer is:
\[
(x, y) = \left(-\frac{10}{3}, \frac{2}{3}\right)
\]
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