Problem 88 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ \frac{4 x}{3}=7 $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{21}{4} \).

Step 1: Identify the Equation

The equation given is \( \frac{4x}{3} = 7 \). Our goal is to solve this equation for \( x \).

Step 2: Eliminate the Fraction

To eliminate the fraction, multiply both sides of the equation by 3 to get rid of the denominator. This gives us: \[ 3 \times \frac{4x}{3} = 3 \times 7 \] which simplifies to: \[ 4x = 21 \].

Step 3: Isolate the Variable

To solve for \( x \), divide both sides of the equation by 4: \[ \frac{4x}{4} = \frac{21}{4} \] This simplifies to: \[ x = \frac{21}{4} \].

Key Concepts

Solving Equations

Solving an equation is like uncovering a mystery. Our task is to find the value of the unknown variable that makes the equation true. In the case of linear equations, such as our example with fractions, the goal is to simplify and solve systematically.
To solve an equation, follow these steps:

• Visualize the equation as a balance. Any operation you perform on one side, you must also perform on the other side.
• Use inverse operations to simplify each step. For example, if you see addition, think of using subtraction to counteract it.
• Continue this process until the variable is isolated, revealing the solution.

Practicing these steps with different types of linear equations will strengthen your understanding and speed up your solving process.

Fractions in Equations

Fractions can sometimes be intimidating, but they are simply numbers expressing a part of a whole. When you see a fraction in an equation, it's important to simplify for ease of solving.
To handle fractions in equations:

• Identify the denominator. In our example, it's 3. The denominator divides the number above it, the numerator.
• To "clear" the fraction, multiply the entire equation by the denominator. This step eliminates the fraction, allowing you to work with whole numbers instead.
• Remember, balancing is key. Multiply both sides by the denominator, ensuring that the equality holds.

Once you remove fractions, the equation often simplifies to a more familiar form, making it easier to solve.

Variable Isolation

Isolating the variable means getting it alone on one side of the equation, revealing its value. This part of solving equations is crucial because it delivers the answer we're looking for.
To isolate the variable:

• First, simplify both sides of the equation as much as possible.
• Use division or multiplication to "undo" the coefficient (the number multiplying the variable). In our example, after clearing the fraction, we divided both sides by 4, the coefficient of x.
• Perform the same mathematical operations to maintain the balance of the equation.

Isolating the variable gives you a clear solution, like deciphering a puzzle. Practice makes perfect when it comes to ensuring your solution is both correct and elegantly simple.