Certainly! Let’s work through each of the parts step-by-step.
### 5. a. Factor to write an equivalent expression:
[tex]\[
-9x + 15y
\][/tex]
1. Identify the greatest common factor (GCF) of the coefficients -9 and 15, which is 3.
2. Factor out -3 from both terms:
[tex]\[
-9x + 15y = -3(3x - 5y)
\][/tex]
So, the factored expression is:
[tex]\[
-3(3x - 5y)
\][/tex]
### 5. b. Expand to write an equivalent expression:
[tex]\[
-5(7m - 2)
\][/tex]
1. Apply the distributive property by multiplying -5 with each term inside the parentheses:
[tex]\[
-5 \cdot 7m + (-5) \cdot (-2) = -35m + 10
\][/tex]
So, the expanded expression is:
[tex]\[
-35m + 10
\][/tex]
### 6. a. Priya's work is incorrect. Explain the error she made.
Priya's work:
[tex]\[
\begin{array}{c}
-2(4 - 2x) - 3 - 2x \\
-8 - 2x - 3 - 2x \\
-11 - 4x
\end{array}
\][/tex]
Priya incorrectly distributed the -2 to both terms in the parentheses but made an error with the sign for the second term. The correct steps are:
1. Distribute -2 to both terms:
[tex]\[
-2(4 - 2x) = -8 + 4x
\][/tex]
Note that [tex]\(-2 \cdot -2x\)[/tex] should be [tex]\(+4x\)[/tex], not [tex]\(-2x\)[/tex] as Priya wrote.
2. Combine all terms:
[tex]\[
-8 + 4x - 3 - 2x
\][/tex]
Combine the like terms:
[tex]\[
-8 - 3 + 4x - 2x = -11 + 2x
\][/tex]
So, Priya's corrected expression is:
[tex]\[
-11 + 2x
\][/tex]
### 6. b. Write an expression equivalent to [tex]\( -4(x + 3) - 12x + 5 \)[/tex] that only has two terms. Simplify.
1. Distribute -4 across the terms in the parentheses:
[tex]\[
-4(x + 3) = -4x - 12
\][/tex]
2. Combine all terms in the expression:
[tex]\[
-4x - 12 - 12x + 5
\][/tex]
3. Combine the like terms:
[tex]\[
-4x - 12x - 12 + 5 = -16x - 7
\][/tex]
So, the simplified expression is:
[tex]\[
-16x - 7
\][/tex]
