Problem 71 - Algebra for College Students

Question

Either solve the given equation or perform the indicated operation \((s),\) whichever is appropriate. $$\frac{2}{x}-\frac{3}{4}=\frac{1}{2}$$

Step-by-Step Solution

Answer

x = \frac{8}{5}

Step 1: Isolate the variable term

Combine the fractions on the left-hand side by eliminating the fraction on the right-hand side. To do this, find a common denominator for the fractions on the left side. The denominators are 4 and 2, and the least common multiple is 4. Multiply every term by 4 to clear the denominators.

Step 2: Multiply all terms by the least common multiple (LCM)

Multiply every term in the equation by 4:\[ 4 \times \frac{2}{x} - 4 \times \frac{3}{4} = 4 \times \frac{1}{2} \]Simplify each term:\[ \frac{8}{x} - 3 = 2 \]

Step 3: Isolate the variable term

Move the constant term to the right side of the equation to isolate the variable term.\[ \frac{8}{x} - 3 + 3 = 2 + 3 \]This simplifies to:\[ \frac{8}{x} = 5 \]

Step 4: Solve for the variable

Now solve for the variable x by cross-multiplying to get rid of the fraction:\[ 8 = 5x \]Divide both sides by 5:\[ x = \frac{8}{5} \]

Key Concepts

least common multiple

The least common multiple (LCM) is crucial when working with rational equations. The LCM is the smallest number that can be evenly divided by each denominator in an equation. In our example equation \(\frac{2}{x} - \frac{3}{4} = \frac{1}{2}\), the LCM of 4 and 2 is 4, because 4 is the smallest number that both 2 and 4 can divide into without leaving a remainder. Multiplying every term by the LCM (which is 4 in this case) eliminates the fractions, making it easier to solve the equation.

isolating variables

Isolating variables means rearranging the equation so that the variable you're solving for is on one side of the equation and everything else is on the other side. This is a key step in solving any equation, including rational equations. After multiplying everything by the LCM and combining like terms, we move the constants to the other side of the equation. For example, in \(\frac{8}{x} - 3 = 2\), we can isolate \( \frac{8}{x} \) by adding 3 to both sides, resulting in \( \frac{8}{x} = 5 \). This leaves the variable term by itself, making it simpler to solve for x.

cross-multiplication

Cross-multiplication is a technique used to solve equations involving fractions. When you have an equation of the form \( \frac{a}{b} = \frac{c}{d} \), you can cross-multiply to get \( a \times d = b \times c \). This process helps rid the equation of fractions, making it easier to solve. In our exercise, after isolating \( \frac{8}{x} = 5 \), cross-multiplication gets rid of the fraction by multiplying both sides by x, resulting in \( 8 = 5x \). From there, it’s straightforward to solve for x.

solving linear equations

Once we've isolated the variable term and applied cross-multiplication, we often end up with a simple linear equation. Linear equations are equations of the first degree, meaning they have no exponents greater than 1. They typically take the form \( ax + b = 0 \). In our example, after cross-multiplying, we got \( 8 = 5x \). Solving this involves simple division; divide both sides by 5 to isolate x: \( x = \frac{8}{5} \). This works out to \( x = 1.6 \), which is the solution to the original equation.