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)
left parenthesis Start Fraction 2 over 3 End Fraction comma negative 3 Start Fraction 1 over 3 End Fraction right parenthesis
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Answer 1

To solve the system of equations using the substitution method, we have the equations:

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

We'll first substitute the expression for \( x \) from the first equation into the second equation.

Substituting \( x = y - 4 \) into the second equation:

\[ (y - 4) + 8y = 2 \]

Now, simplify the equation:

\[ y - 4 + 8y = 2 \] \[ 9y - 4 = 2 \]

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 find \( x \) using \( x = y - 4 \):

\[ x = \frac{2}{3} - 4 \] \[ x = \frac{2}{3} - \frac{12}{3} = \frac{2 - 12}{3} = \frac{-10}{3} \]

So, the solution for the system is:

\[ \left( -\frac{10}{3}, \frac{2}{3} \right) \]

Therefore, the response \( (−\frac{10}{3}, \frac{2}{3}) \) is not one of the options presented. The correct substitution yields the coordinates \( \left( -\frac{10}{3}, \frac{2}{3} \right) \).

Please verify the option formats or check if there was an error. The answer might not match the provided options directly.