To solve the system of equations using substitution, we will first solve one of the equations for one variable and then substitute that expression into the other equation.
Given the equations:
- \( x - 3y = 8 \) (Equation 1)
- \( 4x - 7y = 17 \) (Equation 2)
Step 1: Solve Equation 1 for \( x \)
From Equation 1:
\[ x = 3y + 8 \]
Step 2: Substitute \( x \) in Equation 2
Now we will substitute \( x \) in Equation 2:
\[ 4(3y + 8) - 7y = 17 \]
Step 3: Simplify and solve for \( y \)
Distributing the 4:
\[ 12y + 32 - 7y = 17 \]
Combine like terms:
\[ 12y - 7y + 32 = 17 \] \[ 5y + 32 = 17 \]
Now, subtract 32 from both sides:
\[ 5y = 17 - 32 \] \[ 5y = -15 \]
Divide by 5:
\[ y = -3 \]
Step 4: Substitute \( y \) back into the expression for \( x \)
Now that we have \( y \), we substitute it back into the expression for \( x \):
\[ x = 3(-3) + 8 \]
Calculate:
\[ x = -9 + 8 \] \[ x = -1 \]
Solution
The solution to the system of equations is:
\[ (x, y) = (-1, -3) \]
So the final answer is:
\[ (-1, -3) \]
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