The justification for the step taken from line 2 to line 3 in the given equation is: combining like terms on one side of the equation.
Here's the detailed, step-by-step solution:
1. Start with the original equation:
[tex]\[
3x + 9 - 7x = x + 10 + x
\][/tex]
2. Combine like terms on both sides to simplify the expression.
- On the left-hand side, combine the [tex]\(3x\)[/tex] and [tex]\(-7x\)[/tex]:
[tex]\[
3x - 7x + 9 = -4x + 9
\][/tex]
- On the right-hand side, combine the two [tex]\(x\)[/tex] terms:
[tex]\[
x + x + 10 = 2x + 10
\][/tex]
After combining the like terms, the equation becomes:
[tex]\[
-4x + 9 = 2x + 10
\][/tex]
3. Combine like terms on one side of the equation. This step involves collecting all the [tex]\(x\)[/tex] terms on one side and the constant terms on the other side.
- Subtract [tex]\(2x\)[/tex] from both sides to get all [tex]\(x\)[/tex] terms on one side:
[tex]\[
-4x - 2x + 9 = 2x - 2x + 10
\][/tex]
Simplifying gives:
[tex]\[
-6x + 9 = 10
\][/tex]
4. Isolate the [tex]\(x\)[/tex] term. Subtract 9 from both sides to get:
[tex]\[
-6x + 9 - 9 = 10 - 9
\][/tex]
Simplifying gives:
[tex]\[
-6x = 1
\][/tex]
5. Solve for [tex]\(x\)[/tex] by dividing both sides by -6:
[tex]\[
x = -\frac{1}{6}
\][/tex]
Each step follows algebraic principles correctly, and the justification for step 2 to step 3 is indeed the combining like terms on one side of the equation.
