Let's analyze and solve the given equation step by step to find [tex]\( x \)[/tex].
The given equation is:
[tex]\[ x - 7 + 5x = 36 \][/tex]
1. Combine the like terms on the left side:
Combine the [tex]\( x \)[/tex] terms and the [tex]\( 5x \)[/tex] term:
[tex]\[ x + 5x - 7 = 36 \][/tex]
[tex]\[ 6x - 7 = 36 \][/tex]
2. Isolate the term with [tex]\( x \)[/tex] by adding 7 to both sides:
[tex]\[ 6x - 7 + 7 = 36 + 7 \][/tex]
[tex]\[ 6x = 43 \][/tex]
3. Solve for [tex]\( x \)[/tex] by dividing both sides by 6:
[tex]\[ x = \frac{43}{6} \][/tex]
[tex]\[ x \approx 7.17 \][/tex]
So the correct solution for the equation [tex]\( x - 7 + 5x = 36 \)[/tex] is [tex]\( x \approx 7.17 \)[/tex], not [tex]\( x = 6 \)[/tex].
Now, let's examine the choices Karen might have used:
1. Add [tex]\( x + 5x \)[/tex], add 7 to both sides of the equation:
This is the correct method we used to find [tex]\( x \)[/tex]. Therefore, this method should have given her [tex]\( x \approx 7.17 \)[/tex].
2. Add [tex]\( x - 7 + 5x \)[/tex], add 36 to both sides of the equation:
Adding the left side terms together gives [tex]\( 6x - 7 \)[/tex]. Adding 36 to both sides does not isolate [tex]\( x \)[/tex]. This is incorrect.
3. Add -7 and [tex]\( 5x \)[/tex], subtract [tex]\( x \)[/tex] from both sides of the equation:
Adding these terms on the left side makes [tex]\( -7 + 5x \)[/tex]. Subtracting [tex]\( x \)[/tex] from both sides gives [tex]\( 5x - x - 7 = 36 - x \)[/tex]. This is not a useful step for solving the equation. This method would not isolate [tex]\( x \)[/tex].
4. Add [tex]\( x + 5x \)[/tex], subtract 7 from both sides of the equation:
Adding [tex]\( x \)[/tex] and [tex]\( 5x \)[/tex] gives [tex]\( 6x \)[/tex]. The next logical step is to add 7 to both sides (as Karen's steps initially suggest), not subtract. Hence, this won't solve the equation correctly either.
From the analysis, the closest method Karen might have used is add [tex]\( x + 5x \)[/tex], add 7 to both sides of the equation. However, she should have ended up with [tex]\( x = \frac{43}{6} \)[/tex], not [tex]\( x = 6 \)[/tex]. Therefore, it seems Karen must have made an error in the steps following this combination.
