Solve the system of equations below using substitution. You must show all work to earn full credit. y=x−5 4x+2y=14

Answer 1

To solve the system of equations using substitution, we start with the two equations:

1. \( y = x - 5 \) (Equation 1)
2. \( 4x + 2y = 14 \) (Equation 2)

Step 1: Substitute Equation 1 into Equation 2

We will substitute \( y \) from Equation 1 into Equation 2.

Substituting \( y = x - 5 \) into Equation 2, we get:

\[ 4x + 2(x - 5) = 14 \]

Step 2: Simplify the Equation

Now, let's simplify the left side:

\[ 4x + 2(x - 5) = 4x + 2x - 10 \] \[ = 6x - 10 \]

So the equation becomes:

\[ 6x - 10 = 14 \]

Step 3: Solve for \( x \)

Next, we will solve for \( x \):

\[ 6x - 10 + 10 = 14 + 10 \] \[ 6x = 24 \] \[ x = \frac{24}{6} \] \[ x = 4 \]

Step 4: Substitute \( x \) back to find \( y \)

Now that we have \( x = 4 \), we'll substitute this value back into Equation 1 to find \( y \):

\[ y = x - 5 \] \[ y = 4 - 5 \] \[ y = -1 \]

Step 5: State the Solution

The solution to the system of equations is:

\[ (x, y) = (4, -1) \]

Step 6: Check the Solution

It's important to check our solution in the original equations:

1. Check \( y = x - 5 \): \[ -1 = 4 - 5 \quad \text{(True)} \]

2. Check \( 4x + 2y = 14 \): \[ 4(4) + 2(-1) = 16 - 2 = 14 \quad \text{(True)} \]

Since both equations are satisfied, the solution is confirmed.

Final answer: \( (x, y) = (4, -1) \)