To determine who is correct, let's examine both Spencer's and Jeremiah's approaches step-by-step.
Spencer's Approach: Adding [tex]\(4x\)[/tex] to both sides
1. Start with the original equation:
[tex]\[ 6x - 2 = -4x + 2 \][/tex]
2. Add [tex]\(4x\)[/tex] to both sides to eliminate [tex]\(x\)[/tex] from the right side:
[tex]\[ 6x - 2 + 4x = -4x + 4x + 2 \][/tex]
3. Simplify both sides:
[tex]\[ 6x + 4x - 2 = 2 \][/tex]
[tex]\[ 10x - 2 = 2 \][/tex]
4. Move the constant term [tex]\(-2\)[/tex] to the right side by adding 2 to both sides:
[tex]\[ 10x - 2 + 2 = 2 + 2 \][/tex]
[tex]\[ 10x = 4 \][/tex]
5. Finally, solve for [tex]\(x\)[/tex] by dividing both sides by 10:
[tex]\[ x = \frac{4}{10} \][/tex]
[tex]\[ x = \frac{2}{5} \][/tex]
Spencer's approach yields:
[tex]\[ x = \frac{2}{5} \][/tex]
Jeremiah's Approach: Subtracting [tex]\(6x\)[/tex] from both sides
1. Start with the original equation:
[tex]\[ 6x - 2 = -4x + 2 \][/tex]
2. Subtract [tex]\(6x\)[/tex] from both sides to eliminate [tex]\(x\)[/tex] from the left side:
[tex]\[ 6x - 2 - 6x = -4x - 6x + 2 \][/tex]
3. Simplify both sides:
[tex]\[ -2 = -10x + 2 \][/tex]
4. Move the constant term (2) to the left side by subtracting 2 from both sides:
[tex]\[ -2 - 2 = -10x + 2 - 2 \][/tex]
[tex]\[ -4 = -10x \][/tex]
5. Finally, solve for [tex]\(x\)[/tex] by dividing both sides by -10:
[tex]\[ x = \frac{-4}{-10} \][/tex]
[tex]\[ x = \frac{4}{10} \][/tex]
[tex]\[ x = \frac{2}{5} \][/tex]
Jeremiah's approach also yields:
[tex]\[ x = \frac{2}{5} \][/tex]
Therefore, both approaches are correct in solving for [tex]\(x\)[/tex] since they lead to the same solution, [tex]\( x = \frac{2}{5} \)[/tex]. However, Spencer's approach simplifies the equation more directly by quickly combining like terms on one side, making it a bit more straightforward and quicker to arrive at the solution. Thus, according to simplicity and efficiency, Spencer's approach can be considered correct.
