Problem 57 - Intermediate Algebra

Question

Solve each equation. Check each result. See Example 6. $$ \frac{1}{2}(3 y+2)-\frac{5}{8}=\frac{3}{4} y $$

Step-by-Step Solution

Answer

The solution is \( y = -\frac{1}{2} \). Verified correct.

Step 1: Distribute the Fraction

Begin by distributing the fraction on the left side of the equation. Multiply \( \frac{1}{2} \) by each term inside the parenthesis: \( \frac{1}{2}(3y + 2) = \frac{1}{2} imes 3y + \frac{1}{2} imes 2 = \frac{3}{2}y + 1 \).

Step 2: Set Up Equation Without Parentheses

After distributing, rewrite the equation without parentheses: \( \frac{3}{2}y + 1 - \frac{5}{8} = \frac{3}{4}y \).

Step 3: Combine Like Terms

Simplify the left side by combining the constants:\( 1 - \frac{5}{8} = \frac{3}{8} \), so the equation becomes:\( \frac{3}{2}y + \frac{3}{8} = \frac{3}{4}y \).

Step 4: Isolate Variable Term

Subtract \( \frac{3}{4}y \) from both sides to keep \( y \)-terms on one side:\( \frac{3}{2}y - \frac{3}{4}y = -\frac{3}{8} \).

Step 5: Simplify the Equation

Convert \( \frac{3}{2}y \) to a common denominator with \( \frac{3}{4}y \) to subtract:\( \frac{3}{2} = \frac{6}{4} \), so \( \frac{6}{4}y - \frac{3}{4}y = \frac{3}{4}y \). Thus, the equation is:\( \frac{3}{4}y = -\frac{3}{8} \).

Step 6: Solve for y

Multiply both sides by the reciprocal of \( \frac{3}{4} \) to solve for \( y \):\( y = -\frac{3}{8} \times \frac{4}{3} = -\frac{1}{2} \).

Step 7: Verify the Solution

Substitute \( y = -\frac{1}{2} \) back into the original equation to check:Left Side:\( \frac{1}{2}(3(-\frac{1}{2}) + 2) - \frac{5}{8} = \frac{1}{2}(-\frac{3}{2} + 2) - \frac{5}{8} = \frac{1}{2}(\frac{1}{2}) - \frac{5}{8} = \frac{1}{4} - \frac{5}{8} = -\frac{3}{8} \).Right Side:\( \frac{3}{4}(-\frac{1}{2}) = -\frac{3}{8} \).Both sides match, so \( y = -\frac{1}{2} \) is correct.

Key Concepts

Distributing Fractions

When dealing with equations that contain fractions and parentheses, distributing the fraction correctly is key. Distributing means you'll multiply the fraction by each term inside the parentheses. This allows each term to be treated individually without changing the equation's balance. Let's break down the process:

In the equation \( \frac{1}{2}(3y + 2) \), we multiply each term within the parentheses by \( \frac{1}{2} \). Here's how it's done:

• Multiply \( \frac{1}{2} \times 3y \) to get \( \frac{3}{2}y \)
• Multiply \( \frac{1}{2} \times 2 \) to get 1

After distribution, the expression becomes \( \frac{3}{2}y + 1 \). This allows you to clear the parentheses and simplify further, paving the way to combine like terms or to proceed with solving the equation.

Combining Like Terms

Combining like terms is an essential step when simplifying equations. It involves adding or subtracting terms that have the same variable or no variable at all. For this exercise, it's crucial to deal with constants before jumping to terms with variables.

After distributing the fractions, we have \( \frac{3}{2}y + 1 - \frac{5}{8} \) on the left side of the equation. Our task is to combine like terms:

• The constant terms are: 1 and \( -\frac{5}{8} \). To combine these, convert 1 into a fraction with denominator 8, which is \( \frac{8}{8} \).
• Subtract \( \frac{5}{8} \) from \( \frac{8}{8} \) to get \( \frac{3}{8} \).

The expression simplifies to \( \frac{3}{2}y + \frac{3}{8} \). This simplification aids in refocusing the equation to isolate the variable terms.

Isolating Variables

Isolating the variable \( y \) is the next critical step to finding the solution. This involves getting all terms with \( y \) on one side of the equation, and all constants on the other. This can be achieved through careful subtraction and addition steps.

In the equation \( \frac{3}{2}y + \frac{3}{8} = \frac{3}{4}y \):

• Subtract \( \frac{3}{4}y \) from both sides to bring all \( y \)-terms to the left.
• To subtract fractions effectively, ensure they have the same denominators. Convert \( \frac{3}{2}y \) into \( \frac{6}{4}y \).
• Performing \( \frac{6}{4}y - \frac{3}{4}y \), you get \( \frac{3}{4}y \).

The equation now is \( \frac{3}{4}y = -\frac{3}{8} \), where \( y \) is now isolated on one side enabling you to solve for \( y \) easily.

Checking Solutions

Checking your solution is like a safety measure to ensure the answer is correct. Once you solve for the variable, you can substitute it back into the original equation. This ensures that both sides equate properly, confirming the value found is indeed a solution.

In our problem, we solved for \( y \) and found it to be \( -\frac{1}{2} \). To verify:

• Substitute \( y = -\frac{1}{2} \) into the left and right sides of the original equation.
• Calculate the left side: \( \frac{1}{2}(3(-\frac{1}{2}) + 2) - \frac{5}{8} = -\frac{3}{8} \).
• Calculate the right side: \( \frac{3}{4}(-\frac{1}{2}) = -\frac{3}{8} \).

Both sides equal \( -\frac{3}{8} \), confirming that the value \( y = -\frac{1}{2} \) satisfies the equation and is correct.