Solving Simultaneous Equations Using Substitution Quick Check

Question

2 of 52 of 5 Items
Question
Use the substitution method to solve the system x=y−4, x+8y=2.(1 point)
Responses

infinite solutions
infinite solutions

(−313,23)
left parenthesis negative 3 Start Fraction 1 over 3 End Fraction comma Start Fraction 2 over 3 End Fraction right parenthesis

no solution
no solution

(23,−313)

Answer 1

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

1. \( x = y - 4 \)
2. \( x + 8y = 2 \)

First, we will substitute the expression for \( x \) from the first equation into the second equation.

Substituting \( x \) in the second equation: \[ (y - 4) + 8y = 2 \]

Now combine like terms: \[ y - 4 + 8y = 2 \] \[ 9y - 4 = 2 \]

Next, add 4 to both sides: \[ 9y = 6 \]

Now, divide by 9: \[ y = \frac{6}{9} = \frac{2}{3} \]

Now that we have \( y \), we can substitute back to find \( x \): \[ x = y - 4 = \frac{2}{3} - 4 \] \[ x = \frac{2}{3} - \frac{12}{3} = \frac{2 - 12}{3} = \frac{-10}{3} \]

Thus, the solution is: \[ \left( \frac{-10}{3}, \frac{2}{3} \right) \]

Checking the response options:

• \( (-3 \frac{1}{3}, \frac{2}{3}) \) can be converted to an improper fraction: \( (-\frac{10}{3}, \frac{2}{3}) \).

So the correct response is: \[ \left( -\frac{10}{3}, \frac{2}{3} \right) \]

Thus the correct option from your responses is: \[ (-3 \frac{1}{3}, \frac{2}{3}) \]