To solve the system of equations given by
- \( 6y - 7 = x \)
- \( 3x - 8y = -31 \)
we can use substitution. First, let's express \( x \) in terms of \( y \) from the first equation:
\[ x = 6y - 7 \]
Now, we'll substitute this expression for \( x \) into the second equation:
\[ 3(6y - 7) - 8y = -31 \]
Expanding this equation:
\[ 18y - 21 - 8y = -31 \]
Combining like terms:
\[ 10y - 21 = -31 \]
Now, add 21 to both sides:
\[ 10y = -31 + 21 \]
\[ 10y = -10 \]
Now, divide by 10:
\[ y = -1 \]
Now that we have the value of \( y \), we can substitute it back into the first equation to find \( x \):
\[ x = 6(-1) - 7 \]
\[ x = -6 - 7 \]
\[ x = -13 \]
Therefore, the solution to the system of equations is:
\[ (x, y) = (-13, -1) \]
You can verify this solution by plugging \( x \) and \( y \) back into the original equations to confirm that both are satisfied.
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For \( 6y - 7 = x \):
\[ 6(-1) - 7 = -6 - 7 = -13 \quad \text{(satisfied)} \]
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For \( 3x - 8y = -31 \):
\[ 3(-13) - 8(-1) = -39 + 8 = -31 \quad \text{(satisfied)} \]
Thus, the solution \( (x, y) = (-13, -1) \) is indeed correct.