Problem 51 - 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. $$12-6 x=18-7 x$$

Step-by-Step Solution

Answer

The proposed solution of x = -6 is not correct when checked by substituting back into the original equation.

Step 1: Rearrange the Equation

Add \(6x\) to both sides of the equation to keep the equation balanced: \(12-6x+6x = 18-7x+6x\) simplifies to \(12=x+18\).

Step 2: Isolate x

To isolate x, subtract 18 from both sides of the equation: \(12 - 18 = x+18 - 18\) simplifies to \(-6 = x \).

Step 3: Check the Solution

Substitute \(-6\) (proposed solution for x) back into the original equation. The left hand side becomes \(12 - 6*(-6) = 48\) and the right hand side becomes \(18 - 7*(-6) = 60\). Since 48 and 60 are not equal, the solution is incorrect.

Key Concepts

Addition Property of Equality

Understanding the addition property of equality is crucial when solving algebraic equations. This principle states that you can add the same number to both sides of an equation without changing the equation's solutions. For example, if you have an equation like \( a = b \), you can add \( c \) to both sides such that \( a + c = b + c \). This step maintains the balance of the equation.

In the given exercise \(12 - 6x = 18 - 7x\), we apply this property by adding \(6x\) to both sides to eliminate the \(6x\) term from the left-hand side. It's a strategic move that brings us closer to isolating the variable, which is our ultimate goal. This approach also helps to simplify the equation by consolidating like terms, making it easier to find the solution.

Isolate the Variable

To isolate the variable means to get the variable by itself on one side of the equation, with all other terms on the opposite side. This is a vital step when you aim to solve an equation. It brings clarity to the solution and makes it explicit. After utilizing the addition property of equality in the previous step, we are left with \(12 = x + 18\).

Now, to isolate \(x\), we need to remove the \(18\) from the right-hand side. We achieve this by subtracting \(18\) from both sides of the equation, resulting in \(12 - 18 = x\), which simplifies to \(x = -6\). Isolating the variable is like finding a hidden treasure in a chest; once you clear away the excess, the value you’re seeking is revealed.

Equation Solving Steps

The equation solving steps provide a methodical approach to tackling algebraic equations. It starts with simplifying both sides of the equation if needed, followed by using inverse operations to isolate the variable. Throughout the process, each move we make is governed by the rules of algebra to ensure that we maintain the equation's integrity.

Let's summarize the steps used in our example:

• Step 1: Rearrange the equation and use the addition property of equality to combine like terms. In the exercise, this meant adding \(6x\) to both sides.
• Step 2: Isolate the variable by performing an operation that undoes the action on the variable. We subtracted \(18\) from both sides to isolate \(x\).
• Step 3: Check your solution by substituting it back into the original equation and verify that both sides are equal. In this case, substituting \(x = -6\) did not produce the same number on both sides, indicating an error in the solution process.

Each of these steps is equally important and skipping any can lead to incorrect answers. It's like a roadmap where each signpost guides you to your destination—the correct solution.