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

Responses

(−3 1/3, 2/3)
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

infinite solutions
infinite solutions

(2/3, −3 1/3)

3 answers

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

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

We can 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 it back into the first equation to find \( x \):

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

Thus, the solution to the system of equations is:

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

Now, none of the provided options match this result. Therefore, let's reevaluate the original system to ensure we did not make any mistakes.

  1. From the system:
    • \(x = y - 4\)
    • \(x + 8y = 2\)

We substitute again:

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

This simplifies to:

\[ y - 4 + 8y = 2 \implies 9y - 4 = 2 \implies 9y = 6 \implies y = \frac{2}{3} \]

And substituting back we get:

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

After careful review, the final calculated solution is \( \left( -\frac{10}{3}, \frac{2}{3} \right) \). However, since the processes and calculations were consistent without errors, the choice of options does not include this calculated result.

The options "no solution" and "infinite solutions" are not applicable since we found a unique solution. Therefore, it seems there may have been a misunderstanding in the initial options or problem interpretation.

So, the correct answer would be: \[ \left( -\frac{10}{3}, \frac{2}{3} \right) \] but this is not provided in the choices. If required to choose from the given, we cannot select any option correctly.

one of them is correct.

Let’s go through the process again carefully to identify the correct solution and match it with your given options.

We have the following system of equations:

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

Step 1: Substitute the first equation into the second equation.

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

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

Step 2: Simplify the equation.

Combine like terms:

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

Step 3: Solve for \( y \).

Add 4 to both sides:

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

Step 4: Now substitute \( y \) back to find \( x \).

Using the first equation to get \( x \):

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

Final Solution:

So, we find \( (x, y) = \left( -\frac{10}{3}, \frac{2}{3} \right) \).

Now, let’s check the options you provided:

  1. \( \left(-3 \frac{1}{3}, \frac{2}{3}\right) \)
  2. No solution
  3. Infinite solutions
  4. \( \left(\frac{2}{3}, -3 \frac{1}{3}\right) \)

Here’s how to check \( -3 \frac{1}{3} \):

\[ -3 \frac{1}{3} = -\frac{10}{3} \] So it matches with \( -\frac{10}{3} \).

The corresponding \( y \) value is \( \frac{2}{3} \), which we already calculated.

Thus, the correct solution is:

\[ \left( -3 \frac{1}{3}, \frac{2}{3} \right) \]

This matches option 1. Thank you for your patience!