Find the solution to the system by the substitution method. Check your answer. \[ \begin{array}{ll}-x+3 y=-17 & \text { (1) } \\ 4 x-y=24 & \text { (2) }\end{array} \]

Certainly! Let's solve the given system of equations using the **substitution method** and then verify the solution. \[ \begin{cases} -\,x + 3y = -17 \quad \text{(1)} \\ 4x - y = 24 \quad \text{(2)} \end{cases} \] ### Step 1: Solve one equation for one variable Let's solve **Equation (1)** for \( x \): \[ -\,x + 3y = -17 \] Add \( x \) to both sides: \[ 3y = x - 17 \] Now, solve for \( x \): \[ x = 3y + 17 \quad \text{(3)} \] ### Step 2: Substitute into the other equation Now, substitute \( x = 3y + 17 \) from **Equation (3)** into **Equation (2)**: \[ 4x - y = 24 \] Substituting: \[ 4(3y + 17) - y = 24 \] Expand the multiplication: \[ 12y + 68 - y = 24 \] Combine like terms: \[ 11y + 68 = 24 \] Subtract 68 from both sides: \[ 11y = 24 - 68 \] \[ 11y = -44 \] Divide both sides by 11: \[ y = -4 \] ### Step 3: Find the other variable Now that we have \( y = -4 \), substitute this back into **Equation (3)** to find \( x \): \[ x = 3y + 17 \] \[ x = 3(-4) + 17 \] \[ x = -12 + 17 \] \[ x = 5 \] ### Solution \[ x = 5, \quad y = -4 \] ### Step 4: Check the solution Let's verify the solution by substituting \( x = 5 \) and \( y = -4 \) back into both original equations. **Check in Equation (1):** \[ -\,x + 3y = -17 \] \[ -5 + 3(-4) = -17 \] \[ -5 - 12 = -17 \] \[ -17 = -17 \quad \text{✓} \] **Check in Equation (2):** \[ 4x - y = 24 \] \[ 4(5) - (-4) = 24 \] \[ 20 + 4 = 24 \] \[ 24 = 24 \quad \text{✓} \] Both equations are satisfied, so the solution is correct. ### Final Answer \[ \boxed{\,x = 5,\quad y = -4\,} \]