Problem 99 - Introductory Algebra for College Students

Question

Determine whether the given number is a solution of the equation. $$\frac{1}{3}(x-2)=\frac{1}{5}(x+4)+3 ; 26$$

Step-by-Step Solution

Answer

Yes, -4 is a solution to the equation.

Step 1: Identify the equation and the number

The given equation is \( -7y+18 = -10y+6 \) and the number to check is \( -4 \). We now replace \( y \) in the given equation with \( -4 \) and check if the equation is balanced.

Step 2: Substitute y with -4

Substitute \( y \) with \( -4 \) in the equation: \(-7(-4)+18 = -10(-4)+6 \). This simplifies to \( 28+18 = 40+6 \) which further simplifies to \( 46 = 46 \).

Step 3: Result

Since both sides of the equation are equal, \( -4 \) is a solution to the equation.

Key Concepts

Understanding Equation Balance

Equation balance is a fundamental concept when solving equations. To "balance" an equation means to ensure both sides of it are equal. In algebra, this is crucial because whatever we do to one side of the equation, we must do to the other.

Imagine a balance scale: one side might have weights, and the other side has different weights. To make the scale even, both sides must hold the same weight, symbolizing the equality in a math equation.

When substituting a number into an equation, checking balance involves:

• Plugging the value into the original equation.
• Simplifying both sides of the equation separately.
• Comparing the results to see if both sides match.

This comparison shows if the number is indeed a solution, as seen with the number -4 in the original exercise. If the left and right sides of the equation are equal, the number is a solution.

Substitution Method Explained

The substitution method is a common technique used to solve equations. It's particularly helpful when you want to test a specific value in an equation.

The process is straightforward and involves a few simple steps:

• Identify the variable in the equation that you want to substitute.
• Replace this variable with the given number.
• Simplify both sides of the equation.

For example, in the solution to the exercise provided, we substitute \( y\) with -4 into the equation \(-7y + 18 = -10y + 6\).

Post-substitution, simplification leads us to compare the results—if both sides of the equation end up being equal, it proves our substitution was correct, and therefore, the number is a valid solution.

This method of substitution is essential in confirming specifics, especially when working with linear equations.

Basics of Linear Equations

Linear equations are a type of algebraic equation where each term is either a constant or the product of a constant and a single variable. These equations produce a straight line when graphed, hence the name "linear."

In the context of our exercise, the equation \(-7y + 18 = -10y + 6\) is linear because each term is either a constant or a constant times the variable \( y\).

Main characteristics of linear equations include:

• A degree of 1 (highest power of the variable).
• No variable within exponents, denominators, or under radicals.
• The graph is always a straight line.

Understanding linear equations is foundational for learning more complex algebra concepts. Recognizing their structure helps when applying techniques like the substitution method and ensuring equation balance.

They often appear simple, but mastering these types of equations paves the way for advancing in mathematics.