Problem 44 - Algebra 1

Question

Solve the equation. $$\frac{1}{3} y=5 \frac{2}{3}$$

Step-by-Step Solution

Answer

The solution to the equation is \(y = 17\)

Step 1: Simplify the right side

First thing to simplify the right side of the equation. Express 5 and \(2/3\) as improper fraction. Recall that to convert a mixed number to an improper fraction, multiply the whole number by the denominator, then add the numerator. Here we multiply 5 (the whole number) by 3 (the denominator), then add 2 (the numerator), so the result is \(17/3\). The simplified equation is \(\frac{1}{3}y = \frac{17}{3}\)

Step 2: Solve for y

Now, in order to isolate the variable y, multiply both sides of equation by 3 (not \(\frac{1}{3}\)). That means \(\frac{1}{3}*3y = \frac{17}{3}*3\). Thus, the equation simplifies to \(y = 17\)

Step 3: Confirm the solution

To make sure the solution is correct, substitute y with 17 in the original equation. \(\frac{1}{3}*17 = 5\frac{2}{3}\) which calulates to \(5\frac{2}{3}\); thus, the solution is correct as both sides of the equation are equal.

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and arithmetic operations. In the context of solving linear equations, algebraic expressions are essential because they represent relationships between quantities. For example, in the exercise \( \frac{1}{3} y = 5 \frac{2}{3} \), the expression \( \frac{1}{3} y \) involves a variable \( y \) and multiplication by a fraction, indicating that \( y \) is a third of another number.

When dealing with algebraic expressions such as this, it's crucial to understand how to manipulate them according to the principles of arithmetic and algebra. This includes combining like terms, using the distributive property, and simplifying fractions, among other operations. Mastering the manipulation of algebraic expressions is a fundamental skill that will not only help in solving equations but also in understanding more complex mathematical concepts.

Simplifying Fractions

Simplifying fractions is a critical step in solving algebra equations, especially when they include fractional coefficients or constants. The purpose of simplifying is to make the numbers smaller and easier to work with.

To simplify a fraction, you divide the numerator (top number) and the denominator (bottom number) by their greatest common factor (GCF). However, in the case of converting a mixed number to an improper fraction, as shown in the exercise (\(5 \frac{2}{3}\)), a different approach is needed. You multiply the whole number by the denominator and then add the numerator. This is important because simplifying fractions or converting them to a consistent form can simplify equation solving. Once in an improper fraction form, as with \( \frac{17}{3} \), the expression becomes easier to combine with other fractional terms or to eliminate the fraction entirely when solving for the variable.

Isolate the Variable

To isolate the variable means to get the variable \( y \) by itself on one side of the equation, with everything else on the other side. This is the key step in solving linear equations. The goal is to determine the value of the variable that makes the equation true.

In our exercise, \( \frac{1}{3} y = 5 \frac{2}{3} \) simplifies to the equivalent \( \frac{1}{3} y = \frac{17}{3} \). To isolate \( y \) here, we multiply both sides by the reciprocal of the fraction \( \frac{1}{3} \), which is 3. Multiplying by a reciprocal is a straightforward way to cancel out the fraction. The simplified process looks like this: \( 3 \times \frac{1}{3} y = \frac{17}{3} \times 3 \), leaving \( y = 17 \). Isolating the variable is not only pivotal for finding the solution but also serves as a foundation for solving more complex algebraic equations where several steps may be required to get the variable alone.