Problem 111 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ \frac{2 x}{3}-5=8 $$

Step-by-Step Solution

Answer

The solution is \( x = 19.5 \).

Step 1: Isolate the Term with the Variable

To begin solving the equation \( \frac{2x}{3} - 5 = 8 \), we want to isolate the term with the variable \( x \). Start by adding 5 to both sides of the equation: \[ \frac{2x}{3} - 5 + 5 = 8 + 5 \] This simplifies to: \[ \frac{2x}{3} = 13 \]

Step 2: Eliminate the Fraction

Now, we need to remove the fraction by multiplying both sides by 3, the denominator of the fraction: \[ 3 \times \frac{2x}{3} = 13 \times 3 \] This simplifies to: \[ 2x = 39 \]

Step 3: Solve for the Variable

To solve for \( x \), divide both sides of the equation by 2: \[ \frac{2x}{2} = \frac{39}{2} \] This gives the solution: \[ x = \frac{39}{2} \] which can be written as a decimal: \[ x = 19.5 \]

Key Concepts

Isolating Variables

When solving linear equations, one of the key techniques you'll use is isolating variables. This means getting the variable you are solving for, usually denoted as \( x \), all by itself on one side of the equation. You achieve this by using inverse operations to eliminate other terms.Let's look at an example using the equation \( \frac{2x}{3} - 5 = 8 \). To isolate \( x \), first tackle any addition or subtraction not involving the variable.

• In the example, \( -5 \) is subtracted from \( \frac{2x}{3} \). The inverse operation of subtraction is addition. Therefore, add 5 to both sides to cancel out the \( -5 \).
• After adding 5, the equation simplifies to \( \frac{2x}{3} = 13 \).

Now, the term with the variable is isolated on one side, making it easier to solve for \( x \) in the next steps.

Eliminating Fractions

Fractions can often make equations look complex, but eliminating them is straightforward with multiplication. When your equation has a fraction, multiply every term by the denominator to clear it.Consider our example \( \frac{2x}{3} = 13 \):

• Here, the fraction is \( \frac{2x}{3} \), which has the denominator 3.
• To eliminate the fraction, multiply both sides of the equation by 3.

This eliminates the fraction, simplifying the equation to \( 2x = 39 \). By getting rid of fractions early in the process, the equation becomes easier to manage and prepare for solving the variable in the next step.

Arithmetic Operations

Arithmetic operations, such as addition, subtraction, multiplication, and division, are your fundamental tools when solving equations. They help you carefully balance both sides of the equation to reach a solution.In the simplified equation \( 2x = 39 \), we use division to solve for \( x \):

• To isolate \( x \), divide both sides of the equation by 2, because \( 2 \times x \) means \( x \) is multiplied by 2.
• Division is the inverse operation of multiplication.
• Applying division, you get \( x = \frac{39}{2} \), further simplifying to a decimal: \( x = 19.5 \).

The proper use of arithmetic operations ensures you can manipulate and solve equations accurately, achieving the correct solution for the variable.