Problem 42 - Introductory Algebra for College Students

Question

Solve each equation using the addition property of equality. Be sure to check your proposed solutions. $$x+10.6=-9$$

Step-by-Step Solution

Answer

The solution to the equation \(x + 10.6 = -9\) is \(x = -19.6\).

Step 1: Isolate the Variable

In the given equation \(x + 10.6 = -9\), subtract '10.6' from both sides of the equation to isolate the variable 'x'. This operation will give us \(x + 10.6 - 10.6 = -9 - 10.6\). Simplifying further will result in the expression \(x = -9 - 10.6\).

Step 2: Simplify the Equation

Simplify the right side of the equation by doing the subtraction. This will give us \(x = - 19.6\). This is the proposed solution for 'x'.

Step 3: Verify the Solution

To ensure the solution is correct, substitute 'x' with '-19.6' in the original equation: \(-19.6 + 10.6 = -9\). Simplifying the left side of the equation gives us \(-9\). Since both sides of the equation are equal, the solution is verified.

Key Concepts

Addition Property of Equality

The addition property of equality is a fundamental principle in algebra which states that you can add the same value to both sides of an equation without changing the equation's balance. Imagine a scale in perfect balance. Adding the same weight to both sides maintains this balance. Similarly, when solving the equation \(x + 10.6 = -9\), you're seeking the value of \(x\) that keeps the equation balanced.

To use this principle, you subtract \(10.6\) from both sides to move towards isolating \(x\). The equation then changes to \(x = -9 - 10.6\). By applying this property correctly, we've kept the 'scale' of the equation balanced and moved one step closer to finding the value of \(x\). Remember, subtraction is just the addition of a negative number, so subtracting from both sides is in keeping with this property. Understanding this concept is crucial as it is the basis for most algebraic manipulations.

Isolate the Variable

To solve an algebraic equation, our goal is often to find the value of an unknown variable. 'Isolating the variable' means manipulating the equation to get the variable by itself on one side of the equal sign and a numerical value on the other side. Think of it as the star of the show standing alone on stage under a spotlight. In the provided exercise \(x + 10.6 = -9\), we use the addition property of equality to eliminate the \(10.6\) from the left side,

This gives us \(x = -19.6\) after subtracting \(10.6\) from both sides of the equation. By isolating \(x\), we have made the variable the 'star' and can clearly see its value. Isolating the variable is a technique that will serve students throughout their study of algebra, allowing them to clarify the relationships between different parts of an equation.

Verify Solutions in Algebra

After finding the proposed solution for an algebraic equation, it's not enough to just write it down and move on. You need to plug the solution back into the original equation to verify that it makes the equation true. Verification is the mathematical proof that your solution is correct. Using the original equation \(x + 10.6 = -9\), and replacing \(x\) with the calculated solution \(-19.6\), we see that indeed \(-19.6 + 10.6 = -9\) simplifies to \(-9 = -9\).

Through verification, we've confirmed that the solution maintains the balance of the equation. Students should make a habit of this step to avoid errors and develop a deeper understanding of how different algebraic solutions relate to the original problem. It's the equivalent of cross-checking your answers to ensure accuracy and fully grasp the process of solving equations.