Problem 19 - Algebra A Combined Function

Question

Solve each equation. See Examples 3 through \(5 .\) $$ \frac{3}{4} x-\frac{1}{2}=1 $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 2 \).

Step 1: Isolate the variable term

First, we need to move the constant term to the other side of the equation. We have the equation \( \frac{3}{4} x - \frac{1}{2} = 1 \). To isolate \( \frac{3}{4}x \), add \( \frac{1}{2} \) to both sides to obtain: \[ \frac{3}{4} x = 1 + \frac{1}{2} \] which simplifies to: \[ \frac{3}{4} x = \frac{3}{2} \].

Step 2: Solve for x

Now, divide both sides of the equation by \( \frac{3}{4} \) to solve for \( x \). Dividing by a fraction is equivalent to multiplying by its reciprocal. Thus: \[ x = \frac{3}{2} \times \frac{4}{3} \] This simplifies to: \[ x = 2 \].

Key Concepts

Solving Linear Equations

Linear equations are like puzzles. You have numbers, variables, and operations, and your task is to find the value of the variable. Think of it as balancing scales. Whatever you do to one side of the equation, you must do to the other.
In the problem \( \frac{3}{4} x - \frac{1}{2} = 1 \), your goal is to solve for \( x \). This means making \( x \) the subject of the formula with everything else moved to the opposite side. Start by rearranging terms to get all variable terms on one side and constants on the other. In more complex scenarios, linear equations might involve parentheses or need to be simplified first, but the core idea remains the same.
When solving these equations, keep operations consistent, and always ensure that each step maintains the equation's balance. With practice, you'll solve them as easily as a basic math operation.

Fraction Operations

Dealing with fractions within equations can be tricky at first, but it's all about knowing the rules. Fractions have numerators and denominators, and knowing how to manipulate them is key.
In the given equation, you see the fraction \( \frac{3}{4} x \) and \( \frac{1}{2} \). Adding fractions like \( 1 + \frac{1}{2} \) involves converting whole numbers into fractions or finding a common denominator, making addition straightforward.
To solve for \( x \), you also need to "undo" the multiplication by a fraction. This is where understanding multiplication and division with fractions is crucial. Remember this handy tip: Dividing by a fraction is the same as multiplying by its reciprocal. So, when you divide \( \frac{3}{2} \) by \( \frac{3}{4} \), you're actually multiplying by \( \frac{4}{3} \).

• To divide by \( \frac{a}{b} \), multiply by \( \frac{b}{a} \).
• Make sure to simplify fractions when possible for the neatest result.

Isolating Variables

To solve any equation means to isolate the variable, which is the unknown you need to find. Here, the variable is \( x \), and our aim is to see \( x = ? \) with a clear number as the answer.
Starting with the equation \( \frac{3}{4} x - \frac{1}{2} = 1 \), isolating \( x \) involves getting rid of everything attached to it. This begins with dealing with the subtraction by adding \( \frac{1}{2} \) to both sides. This gives \( \frac{3}{4} x = \frac{3}{2} \).
The next step is eliminating the fraction connected directly to \( x \). You do this by dividing both sides by \( \frac{3}{4} \), effectively isolating \( x \). Each action performed must keep \( x \) as the standalone entity on one side.
These steps not only help solve the equation but also boost your understanding of balancing equations and operations. Isolating the variable is like peeling away layers until only \( x \) remains.