Problem 89 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ \frac{2 x}{5}+2=8 $$

Step-by-Step Solution

Answer

The solution is \( x = 15 \).

Step 1: Isolate the Fraction Term

The goal is to isolate the term containing the variable. Start by subtracting 2 from both sides of the equation to eliminate the constant from the left side.\[ \frac{2x}{5} + 2 - 2 = 8 - 2 \] Simplifying gives:\[ \frac{2x}{5} = 6 \]

Step 2: Eliminate the Denominator

To eliminate the fraction, multiply both sides of the equation by 5, which is the denominator.\[ 5 \times \frac{2x}{5} = 5 \times 6 \]This simplifies to:\[ 2x = 30 \]

Step 3: Solve for x

The final step is to solve for the variable \( x \). Divide both sides by 2 to isolate \( x \):\[ \frac{2x}{2} = \frac{30}{2} \] Simplifying gives:\[ x = 15 \]

Key Concepts

Isolating Variables

When solving linear equations, the first crucial step is isolating the variable. This means we want the variable to stand alone on one side of the equation. By focusing on the side where the variable appears, we aim to simplify the expression by removing other terms. In many cases, such as in the equation \( \frac{2x}{5} + 2 = 8 \), you can achieve this by performing operations like addition or subtraction to both sides. For instance, we subtracted 2 from both sides in our exercise:

• Subtract 2 from both sides: \( \frac{2x}{5} + 2 - 2 = 8 - 2 \)
• Simplifies to: \( \frac{2x}{5} = 6 \)

These operations help simplify the equation and bring us closer to isolating the variable, setting the stage for further steps.

Fraction Elimination

Sometimes, equations include fractions, which can make things a bit more complex. Eliminating fractions is a common task in solving equations and can be easily achieved by clearing the denominator.To remove the fraction in our exercise, we identified the denominator, which is 5, and multiplied every term by it:

• Multiply both sides by 5: \( 5 \times \frac{2x}{5} = 5 \times 6 \)
• Results in: \( 2x = 30 \)

This technique clears the fraction and transforms the equation into an easier form to handle. Removing fractions from the equation simplifies and allows us to focus directly on the variable terms.

Algebraic Manipulation

Once fractions are eliminated, algebraic manipulation steps in to further simplify and solve the equation. Algebraic manipulation involves using basic operations such as addition, subtraction, multiplication, or division to isolate the variable completely on one side.In the final step of the exercise, we used division to isolate \( x \):

• Divide both sides by 2: \( \frac{2x}{2} = \frac{30}{2} \)
• Finally, simplify: \( x = 15 \)

By systematically applying these operations, you can isolate the variable, thus reaching the solution to the equation. It's like peeling away layers to get to the core, where the variable stands alone with its value revealed.