Question

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

infinite solutions
infinite solutions

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

no solution
no solution

(−313,23)

Answer 1

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

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

We will substitute \( x \) from the first equation into the second equation.

**Step 1: Substitute**

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

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

**Step 2: Simplify the equation**

Now, simplify the equation:

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

Combine like terms:

\[
9y - 4 = 2
\]

**Step 3: Solve for \( y \)**

Add 4 to both sides:

\[
9y = 6
\]

Now, divide by 9:

\[
y = \frac{6}{9} = \frac{2}{3}
\]

**Step 4: Find \( x \)**

Now substitute \( y = \frac{2}{3} \) back into the first equation to find \( x \):

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

So, we have:

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

**Step 5: Check the solution**

To check, we can plug \( x = -\frac{10}{3} \) and \( y = \frac{2}{3} \) back into the second equation:

\[
-\frac{10}{3} + 8 \left(\frac{2}{3}\right) = -\frac{10}{3} + \frac{16}{3} = \frac{6}{3} = 2
\]

Both equations are satisfied, so the solution is correct.

**Final Answer: \( \left(-\frac{10}{3}, \frac{2}{3}\right) \)**, which matches the coordinates \( (-\frac{10}{3}, \frac{2}{3}) \) not listed in the responses you provided. However, since none of the options given in your responses (infinite solutions, etc.) seem to relate closely to our calculations, the closest option was likely unlisted.
Please make sure to double-check the options or if the calculation might be rephrased or modified in context to match your options.