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Chapter 2: Problem 26

For each formula, express y as a function of x. $$y-3=\frac{1}{3}(x-4)$$

Short Answer

Expert verified
y = \(\frac{1}{3}x + \frac{5}{3}\)

Step by step solution

01

- Distribute the Fraction

Distribute the \(\frac{1}{3}\) on the right side to both terms inside the parentheses: \(y - 3 = \frac{1}{3}x - \frac{1}{3} \times 4\).
02

- Simplify

Simplify the right-hand side by multiplying the fraction: \(y - 3 = \frac{1}{3}x - \frac{4}{3}\).
03

- Isolate y

Add 3 to both sides to isolate \(y\): \(y = \frac{1}{3}x - \frac{4}{3} + 3\).
04

- Combine Like Terms

Combine \(-\frac{4}{3}\) and \(3\). Convert 3 into a fraction with the same denominator: \(3 = \frac{9}{3}\). Now the equation is: \(y = \frac{1}{3}x - \frac{4}{3} + \frac{9}{3}\). Then combine like terms to get: \(y = \frac{1}{3}x + \frac{5}{3}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

algebraic expression
An algebraic expression is a mathematical phrase that includes numbers, variables, and operation symbols. It is used to represent relationships between different values. In the given exercise, the formula involves the expression \( y - 3 = \frac{1}{3}(x - 4) \). This expression includes numbers (3, \( \frac{1}{3} \, and \, 4 \)), a variable (x), and operations (subtraction and multiplication). Recognizing and understanding these components is important as they form the basis of manipulating the equation to find y as a function of x.
distributing fractions
Distributing fractions involves multiplying the fraction by each term inside parentheses. In our exercise, we distribute \( \frac{1}{3} \) across both x and -4:
  • Start by multiplying \( \frac{1}{3} \, times \, x \) to get \( \frac{1}{3}x \).
  • Then multiply \( \frac{1}{3} \, times \, (-4) \) to get \( -\frac{4}{3} \).
This distribution transforms the original equation to \( y - 3 = \frac{1}{3}x - \frac{4}{3} \). Understanding this concept is crucial as it allows you to break down and simplify more complex algebraic expressions.
isolating variables
Isolating variables means rearranging an equation to get one variable alone on one side. In our exercise, we isolate y by moving other terms to the opposite side. Starting with \( y - 3 = \frac{1}{3}x - \frac{4}{3} \), we add 3 to each side:
  • This gives us \( y = \frac{1}{3}x - \frac{4}{3} + 3 \).
  • By performing the same operation on both sides, we maintain the equality of the equation.
Isolating variables is a fundamental step when solving for a particular unknown in an equation.
combining like terms
Combining like terms means adding or subtracting terms that have the same variables raised to the same power. First, convert 3 to a fraction with denominator 3:
  • \( 3 = \frac{9}{3} \)
Then, combine \( -\frac{4}{3} \) and \( \frac{9}{3} \):
  • Adding these gives \( \frac{5}{3} \), since \( -\frac{4}{3} + \frac{9}{3} = \frac{5}{3} \).
This final step transforms our equation to \( y = \frac{1}{3}x + \frac{5}{3} \). Being able to combine like terms is a necessary skill for simplifying expressions and solving equations effectively.

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