Problem 31 - Prealgebra

Question

Solve equation. Check your solution. \(\frac{1}{3} b+8=\frac{1}{2} b-4\)

Step-by-Step Solution

Answer

The solution is \( b = 72 \).

Step 1: Isolate the variable terms

We start by moving the variable terms to one side of the equation. Subtract \( \frac{1}{3}b \) from both sides:\( 8 = \frac{1}{2}b - \frac{1}{3}b - 4 \).

Step 2: Simplify the equation

Combine the terms with \( b \) on one side. We have \( \frac{1}{2}b - \frac{1}{3}b = \frac{3}{6}b - \frac{2}{6}b = \frac{1}{6}b \). Therefore, the equation becomes:\( 8 = \frac{1}{6}b - 4 \).

Step 3: Isolate the variable

Add 4 to both sides to isolate the \( \frac{1}{6}b \) term:\( 8 + 4 = \frac{1}{6}b \)\( 12 = \frac{1}{6}b \).

Step 4: Solve for the variable

Multiply both sides by 6 to solve for \( b \):\( 12 \times 6 = b \)\( b = 72 \).

Step 5: Check the solution

Substitute \( b = 72 \) back into the original equation: \( \frac{1}{3}(72) + 8 = \frac{1}{2}(72) - 4 \).Calculate both sides: Left side is \( 24 + 8 = 32 \).Right side is \( 36 - 4 = 32 \).Since both sides are equal, our solution \( b = 72 \) is correct.

Key Concepts

Solving Linear Equations

Solving linear equations is a fundamental skill in prealgebra. A linear equation is an equation where the highest power of the variable is one. Solving these equations involves finding the value of the variable that makes the equation true. In our exercise, we started with the equation: \( \frac{1}{3} b + 8 = \frac{1}{2} b - 4 \). The key steps in solving any linear equation include:

• Moving all terms involving the variable to one side of the equation.
• Moving constant terms to the other side.
• Lastly, simplifying the equation to solve for the variable.

For instance, in the exercise, by moving \( \frac{1}{3}b \) from the left to the right side, we gather all the variable terms together. This simplification brings us closer to isolating the variable and solving for its value.

Variable Isolation

Isolating the variable is a crucial step when solving linear equations. The goal here is to have the variable alone on one side of the equation, often the left side, to make reading the solution easier. From our example, after moving the terms involving \( b \) to one side, simplifying them gives us: \( 8 = \frac{1}{6}b - 4 \).
The next step in isolating \( b \) is to eliminate any constants from the side with the variable. By adding 4 to both sides, we get: \( 12 = \frac{1}{6}b \).

• We added 4 to both sides because subtracting 4 was modifying the side with the variable, complicating its isolation.
• Finally, to solve for \( b \), multiply both sides by 6, the reciprocal of \( \frac{1}{6} \), yielding \( b = 72 \).

This step concludes the procedure of isolating the variable, helping you find the actual numerical value of \( b \).

Checking Solutions

After solving an equation, it's important to verify that the solution is correct. This process is known as checking solutions. We substitute the value we've found back into the original equation and see if it balances.
For this exercise, by substituting \( b = 72 \) back into: \( \frac{1}{3}(72) + 8 = \frac{1}{2}(72) - 4 \), two sides of the equation should be equal if our solution is correct.

• On the left, \( \frac{1}{3} \times 72 + 8 = 24 + 8 = 32 \).
• On the right, \( \frac{1}{2} \times 72 - 4 = 36 - 4 = 32 \).

Since both sides equal 32, the solution \( b = 72 \) is verified as correct. Always make sure to perform this check to confirm the solution's accuracy, preventing simple calculation errors.