Problem 15 - 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. $$-2=x+14$$

Step-by-Step Solution

Answer

Solving the equation, the value of \(x\) is \(-16\).

Step 1: Isolate the variable in the equation

In order to isolate \(x\), subtract \(14\) from both sides of the equation, keeping the equation balanced. This results in the equation \(-2 - 14 = x + 14 - 14\). Simplifying both sides results in \(-16 = x\).

Step 2: Verify the Found Solution

Substitute \(-16\) for \(x\) in the original equation to verify. \(-2 = -16+14\), simplifies to \(-2 = -2\), confirming that \(-16\) is indeed the correct solution for \(x\).

Key Concepts

Addition Property of Equality

When solving equations, it's crucial to keep the equation balanced. The addition property of equality is fundamental to achieving this. This property states that you can add or subtract the same number from both sides of an equation without changing its balance.
Applying this concept means that if you have an equation like \(-2 = x + 14\), you can subtract \(14\) from both sides. This keeps the equation true and allows you to move towards finding the solution.
The key point to remember here is that whatever operation you perform on one side must be done to the other side as well. This preserves equality and helps in maintaining a clear path to isolating the variable.

Isolate the Variable

Isolating the variable is the process of rewriting the equation to find out what the variable is equal to. This involves performing operations to both sides of the equation, using the addition property of equality.
In this particular example, the goal is to get \(x\) by itself, so we subtract \(14\) from both sides of the equation. Starting with:

• The equation \(-2 = x + 14\)
• We subtract \(14\) from both sides: \(-2 - 14 = x + 14 - 14\)
• This simplifies to \(-16 = x\), clearly presenting \(x\) as \(-16\).

This step is key because it allows us to turn a complex equation into a simpler, more manageable form. Once the variable is isolated, the solution is more apparent and easier to verify in the context of the original equation.

Verify Solutions

After isolating the variable and determining a potential solution, it is important to verify that this solution is correct. Verification involves substituting the found solution back into the original equation to ensure it holds true.
For our example, we determined that \(x = -16\). To verify, substitute \(-16\) back into the original equation \(-2 = x + 14\):

• Substitute \(-16\) for \(x\): \(-2 = -16 + 14\)
• Simplify: \(-2 = -2\)

The equation is true, confirming that our solution \(x = -16\) is correct. This step is crucial because it checks the accuracy of your solution and ensures there are no errors in the process.