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:
\[
(x, y) = \left(-\frac{10}{3}, \frac{2}{3}\right)
\]
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