Question

solve each system by substitution. check your answer.

1. (\begin{cases}y = 4x\\y = 2x + 6end{cases})

Explanation:

Step1: Substitute \( y = 4x \) into \( y = 2x + 6 \)

Since both equations are solved for \( y \), we can set them equal to each other. So we substitute \( y \) in the second equation with \( 4x \) from the first equation.
\( 4x = 2x + 6 \)

Step2: Solve for \( x \)

Subtract \( 2x \) from both sides of the equation to isolate the \( x \)-terms.
\( 4x - 2x = 2x + 6 - 2x \)
\( 2x = 6 \)
Then divide both sides by 2 to solve for \( x \).
\( \frac{2x}{2} = \frac{6}{2} \)
\( x = 3 \)

Step3: Find \( y \) using \( y = 4x \)

Substitute \( x = 3 \) into the equation \( y = 4x \).
\( y = 4\times3 \)
\( y = 12 \)

Step4: Check the solution

Substitute \( x = 3 \) and \( y = 12 \) into both original equations.

• For \( y = 4x \): Left side \( y = 12 \), Right side \( 4\times3 = 12 \). So \( 12 = 12 \), which is true.
• For \( y = 2x + 6 \): Left side \( y = 12 \), Right side \( 2\times3 + 6 = 6 + 6 = 12 \). So \( 12 = 12 \), which is also true.

Answer:

The solution to the system of equations is \( x = 3 \) and \( y = 12 \), or the ordered pair \( (3, 12) \).