Problem 20 - Prealgebra

Question

Solve each equation. $$2 a-3.3=7 a-5.2$$

Step-by-Step Solution

Answer

The solution is \( a = 0.38 \).

Step 1: Isolate the variable terms on one side

To simplify the given equation, start by subtracting \( 7a \) from both sides. This helps in isolating the variable \( a \) on one side:\[2a - 7a - 3.3 = -5.2\] This simplifies to:\[-5a - 3.3 = -5.2\]

Step 2: Isolate the constant terms

To further simplify the equation, add \( 3.3 \) to both sides to move the constant term from the left side:\[-5a = -5.2 + 3.3\] Evaluating the right side gives:\[-5a = -1.9\]

Step 3: Solve for the variable

To solve for \( a \), divide both sides of the equation by \( -5 \):\[a = \frac{-1.9}{-5}\] Calculating the division gives:\[a = 0.38\]

Key Concepts

Solving Equations

When we talk about solving equations, we're looking for the value of the unknown variable that makes the equation true. Think of an equation as a balance beam: whatever you do to one side, you must also do to the other to keep the balance. This is the essence of solving equations. Let's go through some basic steps to understand how to tackle them.

• Recognize the type of equation you're dealing with. In this case, we are looking at a linear equation.
• Linear equations typically have variables raised to the first power, like "2a - 3.3" in our example.
• Start by simplifying both sides, which can include combining like terms or distributing if necessary.

Once you've simplified both sides, you can start making moves to solve for your variable.

Isolating Variables

The key to solving equations is isolating the variable you're solving for, in this case, the variable "a". This means adjusting the equation in a way that "a" stands alone on one side of the equation. Here's a simple method:

• First, get all terms containing the variable on one side of the equation. You can do this by adding or subtracting terms from both sides.
• Next, deal with any constants (numbers not associated with a variable) on the same side as your variable by performing the opposite mathematical operation.
• Finally, handle any coefficients (numbers multiplying your variable) by dividing both sides of the equation by this coefficient. This leaves the variable by itself on one side of the equation.

In essence, you manipulate the equation step by step until the variable is isolated. If you understand how to isolate variables, you're halfway to solving most algebra problems!

Prealgebra Concepts

Prealgebra lays the groundwork for understanding more advanced mathematical concepts, including algebra. In prealgebra, you learn the foundational skills that are crucial for solving equations such as the one in our exercise. Here's why these basics matter:

• Understanding operations: Knowing how addition, subtraction, multiplication, and division work is crucial. These operations form the basis of all algebraic manipulation.
• Properties of equality: In any equation, what you do to one side, you must do to the other. This helps maintain the equality and keep your equation correct throughout your solving process.
• Working with fractions and decimals: Many equations won't have whole numbers, so understanding how to manipulate fractions and decimals is vital.

By mastering these prealgebra concepts, you're building a solid foundation. It makes tackling algebra problems more intuitive and manageable. The more you familiarize yourself with these basic skills, the easier solving linear equations becomes.