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Kelsey solved the system of equations using the substitution method. Determine the step where a mistake was made and the correction that should be made

Line A:8x+4y=12
Line B: y=2x-5

Step 1 Substitute the expression for y in line a for y in line b 8x+4(2x-5)=12
Step 2 solve for x
8x+8x-20=12
16x-20=12
16x=32
2=x
Step 3 substitute x for y in line a 8x+4(2)=12
8x+8=12
8x=4
x=1/2
Step 4 write the solution as an ordered pair (2,1/2)

1 answer

Let's review Kelsey's solution step by step to determine where the mistake was made.

System of Equations:

  1. Line A: \( 8x + 4y = 12 \)
  2. Line B: \( y = 2x - 5 \)

Step 1:

Kelsey substitutes \( y \) from Line B into Line A:

\[ 8x + 4(2x - 5) = 12 \]

This substitution is correct.

Step 2:

Now, Kelsey simplified the equation:

\[ 8x + 8x - 20 = 12 \] \[ 16x - 20 = 12 \] \[ 16x = 32 \]

This part is also correct, leading to:

\[ x = 2 \] (This is where a mistake is introduced in stating \( 2 = x \) instead of \( x = 2 \).)

Step 3:

Substituting \( x = 2 \) back to find \( y \): Substitute \( x = 2 \) into Line B:

\[ y = 2(2) - 5 \] \[ y = 4 - 5 \] \[ y = -1 \]

Kelsey mistakenly substitutes \( x \) back into Line A:

\[ 8(2) + 4y = 12 \] \[ 16 + 4y = 12 \] \[ 4y = 12 - 16 \] \[ 4y = -4 \] \[ y = -1 \]

Correction Needed:

In Step 2, when solving for \( x \), Kelsey improperly notes it as \( 2 = x \) instead of \( x = 2 \). However, the substitution into Line A in Step 3 was correct after finding \( x = 2 \), but instead of writing \( x = 1/2 \), he should have written \( x = 2 \).

Final ordered pair:

The correct ordered pair solution for the system of equations is \( (2, -1) \), not \( (2, 1/2) \).