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

Solution Steps

1. 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 $x = 6$.

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

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.

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

1. Add $-7$ and $5x$, subtract $x$ from both sides of the equation.
2. Add $x + 5x$, add $7$ to both sides of the equation.
3. Add $x - 7 + 5x$, add $36$ to both sides of the equation.
4. Add $x + 5x$, subtract $7$ from both sides of the equation.

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

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

Among the expressions given:

1. $-8$
2. $7a$
3. $3 + 4x$

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