Find the solution to the system by the substitution method. Check your answers. \( \begin{array}{r}7 x+4 y=86 \\ x+2 y=-2\end{array} \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
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To solve the system by substitution, we first solve one equation for one variable. Let's take the second equation \( x + 2y = -2 \) and solve for \( x \): \[ x = -2 - 2y \] Now we can substitute this expression for \( x \) into the first equation \( 7x + 4y = 86 \): \[ 7(-2 - 2y) + 4y = 86 \] Distributing: \[ -14 - 14y + 4y = 86 \] \[ -14 - 10y = 86 \] Next, isolate \( y \): \[ -10y = 86 + 14 \] \[ -10y = 100 \] \[ y = -10 \] Now substitute \( y = -10 \) back into the expression we found for \( x \): \[ x = -2 - 2(-10) \] \[ x = -2 + 20 \] \[ x = 18 \] So, the solution to the system is \( (18, -10) \). Now, let's check the solution by substituting \( x = 18 \) and \( y = -10 \) back into both original equations: 1. For the first equation: \[ 7(18) + 4(-10) = 126 - 40 = 86 \] (checked) 2. For the second equation: \[ 18 + 2(-10) = 18 - 20 = -2 \] (checked) Both equations are satisfied, so the solution \( (18, -10) \) is correct!
