Question

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

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

(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

infinite solutions

Answer 1

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

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

Now we can substitute the expression for \( x \) from the first equation into the second equation.

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

Now simplify the equation:
\[
y - 4 + 8y = 2
\]
\[
9y - 4 = 2
\]
\[
9y = 2 + 4
\]
\[
9y = 6
\]
\[
y = \frac{6}{9} = \frac{2}{3}
\]

Now that we have the value of \( y \), we can substitute it back into the first equation to find \( x \):
\[
x = y - 4
\]
\[
x = \frac{2}{3} - 4
\]
To subtract, we convert \( 4 \) to a fraction with 3 as the denominator:
\[
4 = \frac{12}{3}
\]
Thus:
\[
x = \frac{2}{3} - \frac{12}{3} = -\frac{10}{3}
\]

Now we have the solution:
\[
\left( -\frac{10}{3}, \frac{2}{3} \right)
\]

This does not match any of the given answer choices exactly. However, the first choice looks like an incorrect representation of fractional answers, so let’s check them more carefully.

1. \( (-\frac{10}{3}, \frac{2}{3}) \) simplifies to \( (-3\frac{1}{3}, \frac{2}{3}) \), which approximates \( (-3.33, 0.67) \).

Now, to match it with the given responses:

- The first response \( (-313,23) \) looks like an incorrect formatting of \( (-\frac{10}{3}, \frac{2}{3}) \).
- The second response \( (\frac{2}{3},-313) \) does not match.
- The third \( no solution \) is incorrect since we found a solution.
- The fourth \( infinite solutions \) is also incorrect.

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

That looks closest to the formatting of \( (-313,23) \).

Final answer:
\[
\left( -\frac{10}{3}, \frac{2}{3} \right) \text{ which corresponds to } (-3\frac{1}{3}, \frac{2}{3}).
\]

No direct matching choice listed seems correct, confirming this system has only one solution.