Questions: Karen found that the solution to x-7+5x=36 is x=6. Which of these is the way she found the solution? - add -7 and 5x, subtract x from both sides of the equation - add x+5x, add 7 to both sides of the equation - add x-7+5x, add 36 to both sides of the equation - add x+5x, subtract 7 from both sides of the equation Which of these expressions is a binomial? Expression 1. -8 2. 7a 3. 3+4x 2 and 3 1 and 2 1 3

Karen found that the solution to x-7+5x=36 is x=6. Which of these is the way she found the solution?
- add -7 and 5x, subtract x from both sides of the equation
- add x+5x, add 7 to both sides of the equation
- add x-7+5x, add 36 to both sides of the equation
- add x+5x, subtract 7 from both sides of the equation

Which of these expressions is a binomial?
Expression
1. -8
2. 7a
3. 3+4x
2 and 3
1 and 2
1
3
Transcript text: Karen found that the solution to $x-7+5 x=36$ is $x=6$. Which of thes the way she found the solution? add -7 and $5 x$, subtract $x$ from both sides of the equation add $x+5 x$, add 7 to both sides of the equation add $x-7+5 x$, add 36 to both sides of the equation add $x+5 x$, subtract 7 from both sides of the equation Which of these expressions is a binomial? \begin{tabular}{|l|l|} \hline Expression & \\ \hline 1 & -8 \\ \hline 2 & $7 a$ \\ \hline 3 & $3+4 x$ \\ \hline \end{tabular} 2 and 3 1 and 2 1 3
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Solution

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Solution Steps

Solution Approach
  1. For the first question, we need to determine the correct steps Karen used to solve the equation x7+5x=36x - 7 + 5x = 36. We will evaluate each option to see which one correctly simplifies the equation to find x=6x = 6.

  2. For the second question, we need to identify which expression is a binomial. A binomial is an algebraic expression containing exactly two terms.

Step 1: Solve the Equation

We start with the equation

x7+5x=36. x - 7 + 5x = 36.

Combining like terms, we have

6x7=36. 6x - 7 = 36.

Adding 77 to both sides gives

6x=43. 6x = 43.

Dividing both sides by 66 results in

x=436. x = \frac{43}{6}.

Since xx does not equal 66, Karen's assertion that the solution is x=6x = 6 is incorrect.

Step 2: Evaluate the Options

We need to determine which method Karen used to arrive at her solution. The options provided are:

  1. Add 7-7 and 5x5x, subtract xx from both sides of the equation.
  2. Add x+5xx + 5x, add 77 to both sides of the equation.
  3. Add x7+5xx - 7 + 5x, add 3636 to both sides of the equation.
  4. Add x+5xx + 5x, subtract 77 from both sides of the equation.

The correct method that leads to the solution is option 4:

add x+5x, subtract 7 from both sides of the equation. \text{add } x + 5x, \text{ subtract } 7 \text{ from both sides of the equation.}

Step 3: Identify the Binomial Expression

Among the expressions given:

  1. 8-8
  2. 7a7a
  3. 3+4x3 + 4x

A binomial is defined as an expression containing exactly two terms. The expression 3+4x3 + 4x fits this definition, as it consists of two distinct terms.

Final Answer

The answers to the questions are:

  • The correct solution to the equation is x=436x = \frac{43}{6}.
  • The method Karen used is option 4: add x+5xx + 5x, subtract 77 from both sides of the equation.
  • The binomial expression is 3+4x3 + 4x.

Thus, the final answers are:

x=436 \boxed{x = \frac{43}{6}}

Correct method: Option 4 \text{Correct method: Option 4}

Binomial expression: 3+4x \text{Binomial expression: } 3 + 4x

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