Problem 45 - Introductory Algebra for College Students

Question

Solve each equation in using both the addition and multiplication properties of equality. Check proposed solutions. $$-5 x=-2 x-12$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 4\).

Step 1: Combine like terms

Rearrange the equation by keeping the terms with variable \(x\) on left side and the numerical term on the right side: \(-5x + 2x = -12\). Simplifying this gives \(-3x = -12\)

Step 2: Solve for x

To find the value of \(x\), divide both sides by -3: \(-3x/-3 = -12/-3\). This simplifies to \(x = 4\)

Step 3: Verification

To verify the solution, substitute the value of \(x\) into the original equation: \(-5*4 = -2*4 - 12\). This simplifies to -20 = -8 - 12. After further simplification, -20 = -20 verifies the solution to be correct.

Key Concepts

Addition and Multiplication Properties of Equality

Understanding the addition and multiplication properties of equality is essential in solving linear equations. These properties demonstrate that you can perform the same mathematical operation, whether it's addition or multiplication, on both sides of an equation without changing the equation's solution.

For example, consider the equation \( -5x = -2x - 12 \) from the exercise. To isolate the variable term on one side, you can add \( 2x \) to both sides of the equation, applying the addition property of equality. This operation gives us \( -5x + 2x = -12 \) or \( -3x = -12 \). It's like saying if two pans of a balance scale are equal, adding the same weight to both pans will still keep them in balance.

Then, by using the multiplication property of equality, you divide both sides of \( -3x = -12 \) by \( -3 \) to solve for \( x \). Since you're performing the same operation on both sides, the equality is preserved, resulting in \( x = 4 \). This step is similar to dividing an entire pie equally among people; everyone gets the same portion.

Combining Like Terms

When you come across terms in an equation that have the same variable raised to the same power, you can combine them to simplify the equation. This process is known as combining like terms and is a powerful tool for solving linear equations efficiently.

Let's look at our exercise where we have \( -5x \) and \( -2x \) on opposite sides of the equation. These are like terms because they both contain the variable \( x \) to the power of one. To combine them, you move \( -2x \) over to the left side by adding \( 2x \) to each side, yielding \( -3x = -12 \) after simplification.

This step of combining like terms is akin to grouping apples together when counting the total number of apples; it just makes things neater and more straightforward. Always make sure to identify and combine like terms early on to simplify your work as much as possible.

Verification of Solutions

After finding a potential solution to an equation, it's crucial to ensure that it actually works. This process, known as verification of solutions, involves substituting the solution back into the original equation to check if it satisfies the equation.

In our example, we determined that \( x = 4 \). To verify this solution, plug \( 4 \) back into the original equation \( -5x = -2x - 12 \), resulting in \( -20 \) on the left side and \( -20 \) on the right side after simplification. Since both sides are equal, the solution is verified. It's like putting a puzzle piece in its place and seeing it fits perfectly.

Verification ensures that you haven't made any mistakes along the way, such as misapplying a property or making an arithmetic error. It's an essential final step to confirm that your solution is indeed correct.