Problem 43 - Introductory Algebra for College Students

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 \(-3.7 + m = -3.7\) is \( m = 0 \).

Step 1: Isolation of Variable 'm'

The question requires solving for 'm'. Therefore, to isolate 'm', add 3.7 to both sides of the equation to offset the initial -3.7 present on the left side. This will provide \( -3.7 + 3.7 + m = -3.7 + 3.7 \). The aim of this step is to cancel out the -3.7 on the left side of the equation as 3.7 - 3.7 equals 0. This operation leaves us with \( m = 0 \).

Step 2: Checking the Proposed Solution

To verify if \( m = 0 \) is a correct solution, substitute '0' for 'm' into the original equation. Therefore, we obtain \( -3.7 + 0 = -3.7 \), which simplifies to \( -3.7 = -3.7 \), proving that the proposed solution is correct.

Key Concepts

Isolation of Variable

To solve an equation, one essential step is isolating the variable, which in our example is 'm'. This means rearranging the equation so that 'm' is alone on one side, ideally the left.

In the given equation \[-3.7 + m = -3.7\],our goal is to get rid of the \(-3.7\)on the left side. By the addition property of equality, what is done to one side must be done to the other to keep both sides equal. Here, we add \(3.7\)to both sides.

• This action cancels out the \(-3.7\)on the left, because \(-3.7 + 3.7 = 0\).
• This leaves us with \(m = 0\), effectively isolating our variable 'm'.

Success in isolating the variable means we're one step closer to solving the equation.

Solving Equations

Once the variable is isolated, solving the equation becomes straightforward. In this scenario, after adding \(3.7\)to both sides, the equation simplifies, and what you are left with is the solution.

Our isolated equation is now \(m = 0\).

• This means the value for 'm' that makes the equation true is \(0\).
• There's nothing more to calculate here since the result is direct.

Always remember:

• Check each side after the variable is isolated to ensure accuracy.
• Ensure the same operation is applied on both sides of the equation.

Verifying Solutions

Verification is a crucial final step that confirms our solution is correct. To do this, we substitute the solved value of the variable back into the original equation.

For our example, replace 'm' with \(0\)in the original equation: \(-3.7 + 0 = -3.7\).

• Both sides equate to \(-3.7\),meaning the solution \(m = 0\)is indeed correct.
• If both sides did not equal, we would know that there was a mistake somewhere in our calculations.

Verification assures us that our approach is reliable and our methods are sound. It’s a key practice to not only find a solution but to confirm it’s the correct one.