Select the correct answer.
What is the justification for step 4 in the solution process?
[tex]\begin{array}{rlrl}
& \frac{9}{2} b+11-\frac{5}{6} b & =b+2 \<
\text { Step 1 } & : & \frac{22}{6} b+11 & =b+2 \<
\text { Step 2 } & & & \frac{8}{3} b+11 & =2 \<
\text { Step 3 } & & & \frac{8}{8} b & =-9 \<
\text { Step 4 } & & & b & =-\frac{27}{8}<
\end{array}[/tex]
A. combining like terms
B. the multiplication property of equality
C. the subtraction property of equality
D. the addition property of equality
Answer (2)
Step 1 combines like terms.
Step 2 should simplify the equation, but there's an error in the provided solution.
Step 3 subtracts 11 from both sides.
Step 4 isolates b using the multiplication property of equality: b = − 8 27 .
Explanation
Analyzing the Problem We are given the equation 2 9 b + 11 − 6 5 b = b + 2 and a series of steps to solve for b . We need to identify the justification for step 4. Let's analyze the steps.
Step 1: Combining Like Terms Step 1 combines the terms with b on the left side of the equation. 2 9 b − 6 5 b = 6 27 b − 6 5 b = 6 22 b . So, the equation becomes 6 22 b + 11 = b + 2 .
Step 2: Simplifying the Fraction Step 2 simplifies the fraction 6 22 to 3 11 . However, the solution states 3 8 b + 11 = b + 2 . This is incorrect. The correct equation should be 3 11 b + 11 = b + 2 . Let's assume the solution meant to have 3 11 b instead of 3 8 b . However, let's continue with the solution as provided.
Step 3: Subtracting 11 from Both Sides Step 3 subtracts 11 from both sides of the equation 3 8 b + 11 = b + 2 . This gives 3 8 b = b − 9 . However, the solution states 3 8 b = − 9 . This is also incorrect. It seems that the b term was dropped. Let's assume that the b term was subtracted from both sides in the step. So, 3 8 b − b = − 9 , which simplifies to 3 5 b = − 9 . However, the solution states 3 8 b = − 9 .
Step 4: Isolating b Step 4 isolates b by multiplying both sides of the equation 3 8 b = − 9 by 8 3 . This gives b = − 9 ⋅ 8 3 = − 8 27 . The justification for this step is the multiplication property of equality.
Justification for Step 4 The justification for step 4 is the multiplication property of equality, where both sides of the equation are multiplied by the same non-zero number to isolate the variable.
Examples
The multiplication property of equality is a fundamental concept in algebra. It's used in various real-world scenarios, such as scaling recipes. If you want to double a recipe, you multiply all the ingredients by 2. This ensures that the ratios of the ingredients remain the same, and the recipe turns out as expected. Similarly, in physics, if you want to calculate the force required to accelerate an object, you use the formula F = ma . If you double the mass m , you need to double the force F to maintain the same acceleration a .
The justification for step 4 is based on the multiplication property of equality, which allows for isolating the variable b . This step is essential for solving the equation. The correct answer is B .
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