Sure, let's go through a detailed, step-by-step solution for solving the equation [tex]\(x - 7 + 5x = 36\)[/tex] to determine which steps Karen might have followed to find [tex]\(x = 6\)[/tex].
### Step-by-Step Solution:
1. Combine like terms on the left side of the equation: The equation is [tex]\(x - 7 + 5x = 36\)[/tex]. We need to combine the terms involving [tex]\(x\)[/tex]:
[tex]\[
x + 5x - 7 = 36
\][/tex]
This simplifies to:
[tex]\[
6x - 7 = 36
\][/tex]
2. Isolate the term with [tex]\(x\)[/tex]: To do this, we need to eliminate [tex]\(-7\)[/tex] from the left side of the equation. We accomplish this by adding 7 to both sides:
[tex]\[
6x - 7 + 7 = 36 + 7
\][/tex]
This simplifies to:
[tex]\[
6x = 43
\][/tex]
3. Solve for [tex]\(x\)[/tex]: To isolate [tex]\(x\)[/tex], we need to divide both sides of the equation by 6:
[tex]\[
x = \frac{43}{6}
\][/tex]
Given the context of the possible steps she might have followed, we see that Karen’s approach has an error since the correct solution should not lead to an integer. However, to match the given possible methods she might have mistakenly thought:
1. Combine [tex]\(x\)[/tex] and [tex]\(5x\)[/tex].
2. Add 7 to both sides of the equation.
So, considering Karen’s point of view:
1. Combine [tex]\(x\)[/tex] and [tex]\(5x\)[/tex]:
[tex]\[
x + 5x
\][/tex]
That gives us:
[tex]\[
6x - 7 = 36
\][/tex]
2. Add 7 to both sides of the equation:
[tex]\[
6x - 7 + 7 = 36 + 7
\][/tex]
This simplifies to:
[tex]\[
6x = 43
\][/tex]
Given this, we can infer that Karen might have followed these specific steps, misinterpreting or simplifying her terms leading to the answer [tex]\(x = 6\)[/tex]:
- Combine [tex]\(x + 5x\)[/tex]
- Add [tex]\(7\)[/tex] to both sides of the equation
Thus, the steps Karen could have followed are:
add [tex]\(x + 5x\)[/tex], add 7 to both sides of the equation.
