Problem 43 - Algebra 1: Concepts and Skills

Question

SOLVING EQUATIONS Multiply by a reciprocal to solve the equation. $$ 12=\frac{2}{3} x $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 18\).

Step 1: Identify the reciprocal of the coefficient

The coefficient of \(x\) is \(\frac{2}{3}\). The reciprocal of \(\frac{2}{3}\) is obtained by switching the numerator and the denominator, that is it would be \(\frac{3}{2}\).

Step 2: Multiply both sides by the reciprocal

Multiply both sides of the equation by the reciprocal, \(\frac{3}{2}\). This gives us: \(12 \times \frac{3}{2}= \frac{2}{3} x \times \frac{3}{2}\).

Step 3: Simplify both sides

Simplify both sides. On the left side, \(12 \times \frac{3}{2} = 18\). On the right side, the fraction \(\frac{2}{3} \times \frac{3}{2}\) simplifies to \(1\), so we're left with \(1 \times x\) or simply \(x\). This gives us: \(18 = x\).

Key Concepts

Reciprocal of a Number

Understanding the concept of a reciprocal is fundamental when solving algebraic equations involving fractions. The reciprocal of a number is simply a fraction flipped upside down. To be clear, it’s the number that, when multiplied by the original, equals to 1.

Consider the number \(\frac{2}{3}\). Its reciprocal is \(\frac{3}{2}\) because \(\frac{2}{3}\times\frac{3}{2}=1\). It’s important to remember that every number, except zero, has a reciprocal. The reciprocal of a whole number, say 5, is its fraction form \(\frac{1}{5}\), because \(5\times\frac{1}{5}=1\).

In the context of solving equations, using the reciprocal allows us to isolate the variable we're solving for by essentially 'cancelling out' the fraction. It's a neat trick that every student should master!

Multiplying Fractions

Multiplying fractions is another essential operation in algebra, and it's simpler than many students think. To multiply two fractions, multiply their numerators (top numbers) and multiply their denominators (bottom numbers) separately.

For example, if we multiply \(\frac{2}{3}\) with its reciprocal \(\frac{3}{2}\), we get \(\frac{2}{3}\times\frac{3}{2}=\frac{2\times3}{3\times2}=\frac{6}{6}=1\). The beauty of multiplying a number by its reciprocal, as shown in this scenario, is that it simplifies to 1, rendering equations easier to solve.

Tips for Multiplying Fractions

• Simplify before multiplying if possible to make your calculation easier.
• Remember that a whole number can be written as a fraction by placing it over 1.
• Always reduce your final answer to its simplest form.

Students can apply these tips to streamline their multiplication process.

Algebraic Equations

An algebraic equation is a statement of equality between two expressions that contain variables. Solving an algebraic equation means finding the value(s) of the variable(s) that makes the equation true.

In the case of the exercise with the equation \(12=\frac{2}{3}x\), our goal is to find the value of \(x\) that satisfies the equation. By multiplying both sides by the reciprocal of \(\frac{2}{3}\), which is \(\frac{3}{2}\), we neatly eliminate the fraction and simplify the equation to a form where \(x\) appears by itself.

Understanding how to handle equations with fractions is crucial for progressing in algebra. Always check your work by plugging the solution back into the original equation to ensure it maintains equality. Being diligent with these practices can help avoid errors and build a stronger foundation in algebra.