Problem 43 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each equation. Using the addition property of equality. Be sure to check your proposed solutions. $$-3.7+m=-3.7$$

Step-by-Step Solution

Answer

The solution to the equation is m = 0.

Step 1: Isolate the variable

In order to isolate 'm', you can add 3.7 to both sides of the equation to counteract the initially subtracted 3.7. This gives: m = -3.7 + 3.7.

Step 2: Simplify the right side of the equation

-3.7 + 3.7 equals 0. Therefore, the simplified equation will be: m = 0.

Step 3: Checking the solution

To verify if the solution is correct, substitute m = 0 into the original equation: -3.7 + 0 = -3.7. The equation holds, so the solution is correct.

Key Concepts

Addition Property of Equality

Understanding the addition property of equality is essential when solving linear equations. In essence, this property states that if you add the same number to both sides of an equation, the equality is still balanced. For example, consider the simple equation \( a = b \). If you add the same number, let’s say \( c \) to both sides, you will then have \( a + c = b + c \), and the equation remains true.

This property is fundamental in algebra and is often employed to isolate variables, a term which we'll discuss in more detail in the respective section. In the exercise \( -3.7 + m = -3.7 \), using this property allows us to add \( 3.7 \) to both sides, eliminating the \( -3.7 \) on the left and helping to isolate \( m \). It's like balancing scales: whatever you do to one side, doing the same to the other side keeps them level.

Isolating the Variable

Isolating the variable is a core part of solving any equation. It means rearranging the equation so that the unknown variable stands alone on one side of the equal sign. The goal is to find the value of this variable. Often, this involves using the addition property of equality, which we've just covered, as well as other operations like subtraction, multiplication, and division, always applied to both sides of the equation to maintain balance.

Let's use our example: to isolate \( m \) in \( -3.7 + m = -3.7 \), we add \( 3.7 \) to both sides, which simplifies to \( m = 0 \). No complicated operations are needed in this case, as \( m \) is already by itself once we add \( 3.7 \) to both sides. Each algebraic step taken to isolate the variable gets us closer to finding the solution.

Checking Solutions in Algebra

Once you've proposed a solution for an equation, it's not time to celebrate just yet! You must check to ensure it’s correct. To do this, substitute your solution back into the original equation. If the equation balances, then you’ve found a correct solution.

In our exercise, we determined the solution \( m = 0 \). To check, we substitute \( 0 \) for \( m \) in the original equation: \( -3.7 + 0 = -3.7 \). It simplifies to \( -3.7 = -3.7 \), which is a true statement, hence confirming our solution is correct. Checking solutions not only confirms accuracy but also reinforces understanding of the concepts at play, such as the addition property of equality and the process of isolating variables.