Questions: Solve the system by substitution. Verify your solution by checking that it satisfies both equations of the system. 3x + 4y = 8 x - 3y = -6

Solution Steps

To solve the system of equations by substitution, we first solve one of the equations for one variable and then substitute this expression into the other equation. Here, we can solve the second equation for \( x \) in terms of \( y \), and then substitute this expression into the first equation to find \( y \). Once \( y \) is found, substitute back to find \( x \). Finally, verify the solution by plugging the values of \( x \) and \( y \) back into both original equations to ensure they hold true.

From the second equation \( x - 3y = -6 \), we can express \( x \) in terms of \( y \): \[ x = 3y - 6 \]

Substituting \( x = 3y - 6 \) into the first equation \( 3x + 4y = 8 \): \[ 3(3y - 6) + 4y = 8 \] This simplifies to: \[ 9y - 18 + 4y = 8 \] Combining like terms gives: \[ 13y - 18 = 8 \]

Adding 18 to both sides: \[ 13y = 26 \] Dividing by 13: \[ y = 2 \]

Now substituting \( y = 2 \) back into the expression for \( x \): \[ x = 3(2) - 6 = 6 - 6 = 0 \]

We need to check if \( (x, y) = (0, 2) \) satisfies both original equations:

1. For \( 3x + 4y = 8 \): \[ 3(0) + 4(2) = 0 + 8 = 8 \quad \text{(True)} \]
2. For \( x - 3y = -6 \): \[ 0 - 3(2) = 0 - 6 = -6 \quad \text{(True)} \]

Since both equations are satisfied, the solution is valid.

Final Answer