Question
Solving Simultaneous Equations Using Substitution Quick Check
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
Skip to navigation
Answers
GPT-4o mini
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.
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.