Problem 16 - Introductory Algebra for College Students

Question

Solve each equation in using the multiplication property of equality. Be sure to check your proposed $$\frac{2}{3} y=12$$

Step-by-Step Solution

Answer

The solution to the equation \(\frac{2}{3} y = 12\) is \(y = 18\).

Step 1: Identify the Reciprocal

First, identify the reciprocal of \(\frac{2}{3}\). The reciprocal of a fraction is simply the result of flipping the fraction over, therefore the reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\).

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

Next, multiply both sides of the equation by \(\frac{3}{2}\) to isolate 'y'. Doing this gives us: \(\frac{2}{3} y * \frac{3}{2} = 12 * \frac{3}{2}\) which simplifies to \(y = 18\)

Step 3: Check the Solution

Finally, substitute the identified value of 'y' into the original equation to check if the proposed solution is correct. When \(y = 18\) is plugged into the original equation \(\frac{2}{3} y = 12\), the left hand side evaluates to 12, which matches the right hand side of the equation. Thus, the solution \(y = 18\) is correct.

Key Concepts

Multiplication Property of Equality

The multiplication property of equality is a fundamental concept in algebra. It states that you can multiply both sides of an equation by the same non-zero number without changing the equation's equality. This is crucial when solving equations, as it allows us to manipulate the equation to isolate variables you want to solve for.

For example, consider an equation like \(\frac{2}{3}y = 12\). To solve for \(y\), you need to perform operations that maintain balance across the equal sign. When you multiply both sides by the reciprocal (a number we’ll explore more in the next section), you ensure that the equality holds. Another point to remember is:

• Always use a non-zero number for multiplication or division.
• Keep the equation balanced by repeating the operation on both sides.

Recognizing the multiplication property of equality aids significantly in preserving the correctness of an equation during the process of isolating the unknown variable.

Reciprocal

In algebra, the reciprocal of a number is an important concept, especially when dealing with fractions. The reciprocal of a fraction \(\frac{a}{b}\) is simply switching the numerator and the denominator, creating \(\frac{b}{a}\).

When solving an equation like \(\frac{2}{3}y = 12\), using the reciprocal can help in simplifying and solving for the variable. To isolate \(y\), multiply both sides by the reciprocal of \(\frac{2}{3}\), which is \(\frac{3}{2}\). This step is vital because:

• The product of a number and its reciprocal is 1 \((\frac{2}{3} * \frac{3}{2} = 1)\).
• Multiplying by the reciprocal cancels out the coefficient in front of the variable.

This technique is particularly useful as fractions and coefficients often appear in equations, and knowing how to adeptly use reciprocals will make solving these equations straightforward.

Checking Solutions

After finding the solution to an equation, it’s vital to check that your derived answer is correct. Checking solutions is the process of substituting the solved value back into the original equation to ensure it satisfies all parts of the equation.

For the equation \(\frac{2}{3}y = 12\) solved earlier, we found \(y = 18\). To check if this solution is correct, substitute \(18\) back into the original equation:
\[\frac{2}{3} * 18 = 12\]
By performing the arithmetic:

• Calculate \(\frac{2}{3} * 18\).
• The result should equal 12, confirming \(y = 18\) is the solution.

Checking your work is an important final step to verify correctness, reduce mistakes, and instill confidence in your answers.