Problem 32 - Prealgebra

Question

Using the addition property of equality first, solve each of the following equations. $$3 x-5=-26$$

Step-by-Step Solution

Answer

The solution is \( x = -7 \).

Step 1: Identify the Equation

The given equation is \( 3x - 5 = -26 \). Our goal is to isolate \( x \) to find its value by using the addition property of equality.

Step 2: Apply the Addition Property

To isolate the term with \( x \), add \( 5 \) to both sides of the equation: \[3x - 5 + 5 = -26 + 5\]This simplifies to:\[3x = -21\]

Step 3: Solve for x

We need to isolate \( x \). Divide both sides of the equation by \( 3 \):\[\frac{3x}{3} = \frac{-21}{3}\]This simplifies to:\[x = -7\]

Key Concepts

Understanding the Addition Property of Equality

The addition property of equality is a fundamental concept in algebra. It states that if you have an equation, adding the same number to both sides of the equation will keep it balanced. Consider it like a seesaw; whatever you add to one side, you must add to the other, ensuring nothing tips over.
For example, in the equation \( 3x - 5 = -26 \), you want to get rid of the \(-5\) so you can focus on isolating \(3x\).
To do this, you add \(5\) to both sides of the equation:

• Left side becomes: \(3x - 5 + 5\) which simplifies to \(3x\).
• Right side becomes: \(-26 + 5\) which simplifies to \(-21\).

This process of adding the same number ensures the equality remains true, effectively moving toward solving for the variable.

Isolation of Variables

Isolation of variables is crucial when solving equations, as the main goal is to determine the value of the unknown variable. Once you've applied the addition property of equality, the next step is to isolate \(x\) on one side of the equation.
Take our simplified equation from the earlier step: \(3x = -21\).
To isolate \(x\), you need to remove the factor that's multiplied with it. In this case, it's \(3\).
Divide both sides of the equation by \(3\) to solve for \(x\):

• Left side: \(\frac{3x}{3}\) which simplifies to \(x\).
• Right side: \(\frac{-21}{3}\) which simplifies to \(-7\).

Now, \(x\) is isolated and equals \(-7\). This crucial step of dividing both sides confirms the solution.

The Essence of Linear Equations

Linear equations represent a basic yet highly important class of equations characterized by linearity: they graph as straight lines on a plane and have no exponents higher than one.
The equation \(3x - 5 = -26\) is a classic example of a linear equation. Here, each term is either a constant or a multiple of the variable \(x\).
Such equations are all about finding that point where the equation holds true, meaning both sides of the equation are equal.
The steps we used, starting from applying the addition property of equality to isolating the variable, are methods focused on maintaining this equality while rearranging the terms.

• They are simple to solve if you consistently apply arithmetic operations.
• Understanding linear equations is foundational for managing more complex algebraic equations later on.

Keep practicing these steps, and you'll smoothly solve any linear equations you come across!