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

Solution

Solution Steps

Solution Approach

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

\[ x - 7 + 5x = 36. \]

Combining like terms, we have

\[ 6x - 7 = 36. \]

Adding $7$ to both sides gives

\[ 6x = 43. \]

Dividing both sides by $6$ results in

\[ x = \frac{43}{6}. \]

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

Step 2: Evaluate the Options

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

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.

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

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

Step 3: Identify the Binomial Expression

Among the expressions given:

$-8$
$7a$
$3 + 4x$

A binomial is defined as an expression containing exactly two terms. The expression $3 + 4x$ 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 $x = \frac{43}{6}$.
The method Karen used is option 4: add $x + 5x$, subtract $7$ from both sides of the equation.
The binomial expression is $3 + 4x$.

Thus, the final answers are:

\[ \boxed{x = \frac{43}{6}} \]

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

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

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