Problem 37 - Algebra 1: Concepts and Skills

Question

SOLVING EQUATIONS Multiply by a reciprocal to solve the equation. $$ \frac{3}{4} k=1 $$

Step-by-Step Solution

Answer

The solution to the equation \(\frac{3}{4} k = 1\) is \(k = \frac{4}{3}\).

Step 1: Identify the Reciprocal of the Coefficient

The coefficient of the variable k is \(\frac{3}{4}\), and its reciprocal is \(\frac{4}{3}\). A reciprocal is found by flipping a fraction; the numerator becomes the denominator and the denominator becomes the numerator.

Step 2: Multiply Both Sides of the Equation by the Reciprocal

Multiply \(\frac{3}{4} k = 1\) on both sides by \(\frac{4}{3}\). This gives \(\frac{4}{3} * \frac{3}{4} k = \frac{4}{3} * 1\). The result is \(k = \frac{4}{3}\).

Step 3: Simplify the Result

The result \(k = \frac{4}{3}\) is already simplified as this equation cannot be further reduced.

Key Concepts

Reciprocal of a Fraction

Understanding the reciprocal of a fraction is fundamental in various areas of mathematics, including the process of solving equations. A reciprocal simply means flipping the numerator (top number) and denominator (bottom number) of a fraction. For example, the reciprocal of \( \frac{2}{7} \) is \( \frac{7}{2} \). This comes in handy when you want to get rid of a fraction in an equation. By multiplying a fraction by its reciprocal, you essentially cancel it out because the product is 1. For example, \( \frac{3}{4} \) multiplied by \( \frac{4}{3} \) equals 1. This property is what allows us to solve equations efficiently by 'clearing' fractions.

Solving Linear Equations

Solving linear equations is a staple of algebra. A linear equation is one where the variable, say \( x \), is raised to the first power. The general solutions follow consistent steps: isolate the variable on one side of the equation, simplify, and solve. When you encounter a coefficient that is a fraction, you can use the reciprocal of that fraction to simplify. In the example \( \frac{3}{4} k = 1 \), you would multiply both sides by the reciprocal of \( \frac{3}{4} \), which is \( \frac{4}{3} \), to isolate \( k \). In fact, solving any linear equation relies on a balance — whatever operation you perform on one side, you must also do to the other, to maintain equality.

Multiplying Fractions

Multiplying fractions might seem complicated at first, but it follows a simple rule: multiply the numerators together and the denominators together. For instance, to multiply \( \frac{1}{2} \) by \( \frac{3}{4} \), you would calculate \( \frac{1 \times 3}{2 \times 4} = \frac{3}{8} \). No common denominator is needed for multiplication, unlike with addition or subtraction of fractions. And remember, multiplying by a reciprocal is special because the numerators and denominators will cancel out, leaving you with 1—this is a powerful tool in solving equations involving fractions.

Algebraic Manipulation

Algebraic manipulation involves rewriting expressions and equations to simplify them or to represent them in a different but equivalent way. It encompasses various operations such as expanding brackets, factorizing, and simplifying expressions. The goal is often to isolate the variable when solving an equation. Techniques such as 'cross multiplication' when dealing with proportions, or 'adding the same amount to both sides' to keep the equality balanced, are part of this arsenal. The crux is understanding the properties of numbers and operations: associative, distributive, commutative, and identity properties all play an essential role in manipulating algebraic expressions effectively.