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Chapter 3: Problem 35

Solve each equation. \(6(6-4 y)=12 y\)

Short Answer

Expert verified
y = 1

Step by step solution

01

Distribute

First, distribute the 6 on the left side of the equation: \(6 \times (6 - 4y) = 12y\). This becomes \(36 - 24y = 12y\).
02

Move Variables to One Side

Add 24y to both sides of the equation to bring all variable terms to one side: \(36 = 12y + 24y\). This simplifies to \(36 = 36y\).
03

Solve for y

Divide both sides by 36 to solve for \(y\): \(y = \frac{36}{36}\). This simplifies to \(y = 1\).

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Key Concepts

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

Distribution in Algebra
In algebra, distribution refers to the process where each term inside a parenthesis is multiplied by a term outside the parenthesis. This is based on the distributive property of mathematics, which states:
  • \( a(b + c) = ab + ac \)
To apply distribution in algebra, you must multiply each term within the parentheses by the term outside. For the given equation, we see this in action:
  • Starting with \( 6(6 - 4y) \), we distribute the 6 to both 6 and \(-4y\).
  • This results in \( 6 \times 6 - 6 \times 4y \). Simplifying, we have \( 36 - 24y \).
By following these steps, the equation is simplified, allowing for easier handling of the terms when moving on to solve the equation.
Solving Linear Equations
Solving linear equations means finding the value of the variable that makes the equation true. Linear equations are of first degree, which means that the variable is only raised to the power of one. The general form is:
  • \( ax + b = c \)
To solve these equations, several steps are usually followed to bring the equation to its simplest form. For instance, in our example:
  • We first distribute any coefficients as mentioned before.
  • Then, we aim to simplify the equation by combining like terms on each side.
The objective is to isolate the variable on one side of the equation to easily determine its value, leading us to the next crucial concept.
Variable Isolation
Variable isolation is a crucial step in solving equations, as it allows you to determine the value of the variable directly. The aim is to have the variable alone on one side of the equation and all constants on the other. Let's see how this applies to our example:
  • Starting with the equation \( 36 - 24y = 12y \), we add \( 24y \) to both sides to get all the terms involving \( y \) on one side: \( 36 = 12y + 24y \).
  • This simplifies to \( 36 = 36y \).
  • We then divide both sides by 36 to isolate \( y \) completely: \( y = \frac{36}{36} \), simplifying to \( y = 1 \).
Through these steps, you ensure that the value of the variable is clearly identified, achieving the solution to the equation. Understanding how to isolate a variable effectively is integral in solving algebraic equations and will aid in tackling more complex problems.

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