Both Spencer and Jeremiah are correct in their approaches to solving the equation [tex]\( 6x - 2 = -4x + 2 \)[/tex]. Let's break down each method step by step:
### Spencer's Approach
Spencer suggests adding [tex]\( 4x \)[/tex] to both sides of the equation:
[tex]\[ 6x - 2 = -4x + 2 \][/tex]
Add [tex]\( 4x \)[/tex] to both sides:
[tex]\[ 6x - 2 + 4x = -4x + 2 + 4x \][/tex]
Simplify the equation:
[tex]\[ 10x - 2 = 2 \][/tex]
Next, add 2 to both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[ 10x - 2 + 2 = 2 + 2 \][/tex]
[tex]\[ 10x = 4 \][/tex]
Finally, divide both sides by 10 to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{4}{10} \][/tex]
[tex]\[ x = \frac{2}{5} \][/tex]
### Jeremiah's Approach
Jeremiah suggests subtracting [tex]\( 6x \)[/tex] from both sides of the equation:
[tex]\[ 6x - 2 = -4x + 2 \][/tex]
Subtract [tex]\( 6x \)[/tex] from both sides:
[tex]\[ 6x - 2 - 6x = -4x + 2 - 6x \][/tex]
Simplify the equation:
[tex]\[ -2 = -10x + 2 \][/tex]
Next, subtract 2 from both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[ -2 - 2 = -10x + 2 - 2 \][/tex]
[tex]\[ -4 = -10x \][/tex]
Finally, divide both sides by -10 to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{-4}{-10} \][/tex]
[tex]\[ x = \frac{4}{10} \][/tex]
[tex]\[ x = \frac{2}{5} \][/tex]
### Conclusion
Both methods lead to the same solution, [tex]\( x = \frac{2}{5} \)[/tex]. Thus, both Spencer and Jeremiah are correct in their initial steps and their overall approach to solving the equation.
