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What is the justification for step 4 in the solution process?

[tex]\[
\begin{aligned}
\text{Step 1:} & \quad \frac{9}{2} b + 11 - \frac{5}{6} b = b + 2 \\
\text{Step 2:} & \quad \frac{22}{6} b + 11 = b + 2 \\
\text{Step 3:} & \quad \frac{8}{3} b + 11 = 2 \\
\text{Step 4:} & \quad \frac{8}{3} b = -9 \\
\text{Step 5:} & \quad b = -\frac{27}{8}
\end{aligned}
\][/tex]

What is the justification for Step 5?

A. the addition property of equality
B. combining like terms
C. the subtraction property of equality
D. the multiplication property of equality

Sagot :

GinnyAnswer
Let's go through the steps provided in the problem statement to determine the justification for Step 4.

We start with the given equation and simplify it through each step:

Step 1:
[tex]\[ \frac{9}{2}b + 11 - \frac{5}{6}b = b + 2 \][/tex]

We rewrite the equation on a common denominator:
[tex]\[ \frac{27}{6}b + 11 - \frac{5}{6}b = b + 2 \][/tex]

Combining the terms involving \( b \):
[tex]\[ \frac{22}{6}b + 11 = b + 2 \][/tex]

This is simplified to:
[tex]\[ \frac{11}{3}b + 11 = b + 2 \][/tex]

Step 2:
Subtract \( b \) from both sides to isolate the terms involving \( b \):
[tex]\[ \frac{8}{3}b + 11 = 2 \][/tex]

Step 3:
Subtract 11 from both sides to further isolate the \( b \) term:
[tex]\[ \frac{8}{3}b = -9 \][/tex]

Step 4:
To solve for \( b \), you need to isolate \( b \) by multiplying both sides of the equation by the reciprocal of \( \frac{8}{3} \), which is \( \frac{3}{8} \):
[tex]\[ b = -9 \times \frac{3}{8} \][/tex]
[tex]\[ b = -\frac{27}{8} \][/tex]

The operation performed in Step 4 is to multiply both sides of the equation by the reciprocal of \( \frac{8}{3} \), which is \( \frac{3}{8} \). This operation is known as the multiplication property of equality.

Thus, the correct justification for Step 4 is:

D. the multiplication property of equality
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