Problem 15 - Prealgebra

Question

Solve each equation. $$\frac{1}{3} x-2.99=1.02$$

Step-by-Step Solution

Answer

The solution is \( x = 12.03 \).

Step 1: Isolate the variable term

The equation is given as \( \frac{1}{3} x - 2.99 = 1.02 \). First, we need to get \( \frac{1}{3} x \) by itself on one side of the equation. To do that, we add 2.99 to both sides:\[ \frac{1}{3} x - 2.99 + 2.99 = 1.02 + 2.99 \]This simplifies to:\[ \frac{1}{3} x = 4.01 \]

Step 2: Solve for the variable

Now that we have \( \frac{1}{3} x = 4.01 \), we need to solve for \( x \). Multiply both sides of the equation by 3 to eliminate the fraction:\[ 3 \cdot \frac{1}{3} x = 3 \cdot 4.01 \]This simplifies to:\[ x = 12.03 \]

Key Concepts

Isolating the Variable

Isolating the variable is a crucial first step in solving linear equations. The main goal is to have the variable, in this case, \( x \), by itself on one side of the equation. This makes it much easier to find its value. We started with the equation \( \frac{1}{3}x - 2.99 = 1.02 \). Notice the term \( -2.99 \) on the same side as \( \frac{1}{3}x \). To isolate \( \frac{1}{3}x \), we need to get rid of \( -2.99 \).
You can do this by performing the opposite operation. Since \( -2.99 \) means we are subtracting 2.99, the opposite operation is to add 2.99 to both sides of the equation. This gives:

• \( \frac{1}{3}x - 2.99 + 2.99 = 1.02 + 2.99 \)

This simplification leads to:

• \( \frac{1}{3}x = 4.01 \)

The term \( \frac{1}{3} x \) is now isolated, ready for the next step. By focusing on isolating the variable, we simplify the equation into a more manageable form.

Fractions in Equations

Solving equations with fractions can seem tricky at first, but with practice, it becomes straightforward. Fractions are just numbers divided by other numbers, so the key is understanding how they interact with other operations. In our example equation, the fraction \( \frac{1}{3}x \) is on one side.The fraction \( \frac{1}{3} \) indicates that \( x \) is divided by 3. To "undo" the division by 3, we multiply both sides of the equation by 3. This step will eliminate the fraction:

• Multiply both sides: \( 3 \cdot \frac{1}{3}x = 3 \cdot 4.01 \)

This simplifies to just \( x \) on the left-hand side, because:

• \( 3 \cdot \frac{1}{3} = 1 \), so \( x = 3 \cdot 4.01 \)

Remember, when dealing with fractions in equations, multiplication or division helps to eliminate them, making equations simpler to solve.

Step-by-Step Solutions

Working through equations step-by-step ensures that no mistakes are made and that each part of the equation is handled correctly. For our original problem:1. **Isolate the Variable**: - Start with \( \frac{1}{3} x - 2.99 = 1.02 \). - Add 2.99 to both sides to isolate \( \frac{1}{3}x \): - Result: \( \frac{1}{3}x = 4.01 \).2. **Solve for the Variable**: - Multiply both sides by 3 to eliminate the fraction: - \( x = 3 \times 4.01 \). - Simplify: - \( x = 12.03 \).By following each step methodically, you ensure that your solution is correct and all operations are properly performed. This process also gives you a clearer understanding of how each component of the equation affects the result. Practicing step-by-step solving enhances problem-solving skills, making future equations easier to tackle.