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Chapter 2: Problem 47

Solve each equation with fraction coefficients. $$ \frac{1}{2} a+\frac{3}{8}=\frac{3}{4} $$

Short Answer

Expert verified
a = \(\frac{3}{4}\)

Step by step solution

01

- Eliminate the Fraction on the Right Side

First, subtract \(\frac{3}{8}\) from both sides to isolate the term with variable \(a\). \[ \frac{1}{2} a + \frac{3}{8} - \frac{3}{8} = \frac{3}{4} - \frac{3}{8} \ \frac{1}{2} a = \frac{3}{4} - \frac{3}{8} \]
02

- Find a Common Denominator

Convert \(\frac{3}{4}\) and \(\frac{3}{8}\) to have a common denominator to subtract them. \[ \frac{3}{4} = \frac{6}{8} \ Now our equation is: \frac{1}{2} a = \frac{6}{8} - \frac{3}{8} \ \frac{1}{2} a = \frac{3}{8} \]
03

- Eliminate the Fraction Coefficient

To solve for \(a\), multiply both sides by the reciprocal of \(\frac{1}{2}\), which is 2. \[ 2 \cdot \frac{1}{2} a = 2 \cdot \frac{3}{8} \ a = \frac{6}{8} \ Simplify if possible. a = \frac{3}{4} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

fraction coefficients
When you see an equation like \( \frac{1}{2} a + \frac{3}{8} = \frac{3}{4} \), it contains what we call 'fraction coefficients'. In this case, the term \( \frac{1}{2} a \). Fraction coefficients can make equations seem hard to solve.
The important thing to remember is to follow proper steps to simplify them. Simplifying involves dealing with fractions directly, and this brings us to the next concept: finding a 'common denominator'.
common denominator
To subtract or add fractions, you need a common denominator.
This is a number that both denominators divide into evenly. In our example, we needed the least common multiple of 4 and 8 for the terms \( \frac{3}{4} \) and \( \frac{3}{8} \).
Since 8 is a multiple of both, it becomes our common denominator. We then convert \( \frac{3}{4} \) to \( \frac{6}{8} \) so both fractions are in terms of eighths. This makes it easier to subtract: \( \frac{6}{8} - \frac{3}{8} \).
After converting, it鈥檚 straightforward to perform the subtraction.
eliminating fractions
Eliminating fractions from equations is crucial for simpler calculations. Once we have \( \frac{1}{2} a = \frac{3}{8} \), we simplify by eliminating the fraction coefficient.
Multiplying both sides by the reciprocal of \( \frac{1}{2} \), which is 2, cancels out the fraction:\( 2 \cdot \frac{1}{2} a = 2 \cdot \frac{3}{8} \).
This simplifies to \( a = \frac{6}{8} \). Finally, we simplify \( \frac{6}{8} \) to \( \frac{3}{4} \). By eliminating the fractions, we make solving the equation much cleaner and more manageable.
step-by-step equation solving
Solving fractional equations is all about taking it step by step. Here鈥檚 a recap of our process:
  • First, isolate the variable term by subtracting fractions on both sides. In our example: \( \frac{1}{2} a = \frac{3}{4} - \frac{3}{8} \).
  • Next, find a common denominator for subtraction: \( \frac{3}{4} \) becomes \( \frac{6}{8} \), making it easier to subtract: \( \frac{6}{8} - \frac{3}{8} = \frac{3}{8} \).
  • Finally, eliminate the fraction coefficient by multiplying both sides by its reciprocal: \( 2 \cdot \frac{1}{2} a = 2 \cdot \frac{3}{8} \) and simplifying the resulting equation.

By following these steps in sequence, even complex fractional equations become much more straightforward to solve.

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Most popular questions from this chapter

Solve. For each inequality, also graph the solution and write the solution in interval notation. $$ |3 x-4| \geq 2 $$

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