Problem 84 - Introductory Algebra for College Students

Question

Solve and check: \(5 x+20=8 x-16\)

Step-by-Step Solution

Answer

The solution to the equation \(5x + 20 = 8x - 16\) is \(x = 12\).

Step 1: Isolate \(x\) terms

Subtract \(5x\) from both sides of the equation to get the \(x\) terms on one side. This gives: \(20 = 8x - 5x -16\). Simplify the right side by combining like terms which results in: \(20 = 3x - 16\).

Step 2: Isolate constant terms

Next, add 16 to both sides of the equation to isolate constant terms on left side. It results in: \(20 + 16 = 3x\). This further simplifies to: \(36 = 3x\).

Step 3: Solve for \(x\)

Finally, divide both sides of the equation by 3 to solve for \(x\). We get: \(x = 36 / 3\). So, \(x = 12\).

Step 4: Checking

To check, substitute \(x = 12\) into the original equation: \(5(12) + 20 = 8(12) - 16\). This simplifies to \(60 + 20 = 96 - 16\), and further to \(80 = 80\), which confirms that \(x = 12\) is the solution.

Key Concepts

Variable Isolation

When solving linear equations, the primary goal is to "isolate" the variable on one side of the equation. This means we want the variable, commonly represented as \(x\), to stand alone, so that we can determine its value easily. To achieve this, we perform operations that help separate \(x\) from other terms. For example, in the exercise \(5x + 20 = 8x - 16\), the initial step involved subtracting \(5x\) from both sides. This operation aligns all the \(x\) terms on one side of the equation, aiding in clarity.To isolate the variable completely, we repeat these steps, balancing both sides of the equation until \(x\) is alone. One must adhere to the principle of performing equal operations on both sides to maintain equality. Once variable isolation is complete, solving the equation becomes a straightforward task.

Combining Like Terms

In algebra, simplifying an equation often involves "combining like terms." This process refers to adding or subtracting terms that have the same variables raised to the same power. In the context of our example, after isolating the \(x\) terms on one side, we saw: \(8x - 5x - 16\).Here, \(8x\) and \(-5x\) are like terms because they each contain the variable \(x\). By combining them, we simplify the equation to \(3x - 16\). Simplifying expressions by combining like terms is vital as it reduces the equation's complexity, making subsequent steps more manageable.This step is essential, as it not only simplifies the equation but also reduces potential confusion, leading to a clearer path to the solution.

Verification of Solutions

After finding a solution, it's crucial to verify if it satisfies the original equation. Verification ensures that no mistakes were made during simplification or arithmetic operations. In our problem, after determining \(x = 12\), we should substitute this value back into the original equation: \(5(12) + 20 = 8(12) - 16\).This substitution should lead to an equality: \(80 = 80\). Checking this step reassures us that our solution of \(x = 12\) is correct. If the two sides of the equation are not equal, it indicates an error in the calculations.Always make verification part of your solving process. This practice helps develop mathematical accuracy and confidence in your solutions.