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Choose the equations below that represent correct steps in solving this system of equations using substitution.

x + 3y = 9
y = x - 5

1 answer

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

  1. \( x + 3y = 9 \)
  2. \( y = x - 5 \)

Since the second equation already expresses \( y \) in terms of \( x \), we can substitute this expression for \( y \) into the first equation.

Here are the correct steps to solve the system:

  1. Substitute \( y = x - 5 \) into the first equation: \[ x + 3(x - 5) = 9 \]

  2. Distribute \( 3 \) across \( (x - 5) \): \[ x + 3x - 15 = 9 \]

  3. Combine like terms: \[ 4x - 15 = 9 \]

  4. Add \( 15 \) to both sides to isolate the term with \( x \): \[ 4x = 24 \]

  5. Divide by \( 4 \) to solve for \( x \): \[ x = 6 \]

  6. Substitute \( x = 6 \) back into the equation for \( y \) (using \( y = x - 5 \)): \[ y = 6 - 5 = 1 \]

Thus, the solution to the system of equations is \( x = 6 \) and \( y = 1 \).

In summary, here are the correct equations reflecting the steps in solving the system using substitution:

  1. \( y = x - 5 \) (given equation)
  2. \( x + 3(x - 5) = 9 \) (substitution into the first equation)
  3. \( 4x - 15 = 9 \) (after distributing)
  4. \( 4x = 24 \) (on adding 15 to both sides)
  5. \( x = 6 \) (after dividing)
  6. \( y = 1 \) (substituting back to find \( y \))

These represent the valid steps taken in the substitution method to solve the system of equations.