Problem 35 - Prealgebra

Question

Simplify each side of the following equations before applying the addition property. $$6 a-2-5 a=-9+1$$

Step-by-Step Solution

Answer

The solution is \(a = -6\).

Step 1: Simplify the Left Side

First, look at the left side of the equation: \(6a - 2 - 5a\). Combine the like terms \(6a\) and \(-5a\). You get: \((6a - 5a) - 2 = 1a - 2\). So, the simplified left side is \(a - 2\).

Step 2: Simplify the Right Side

Now, simplify the right side of the equation: \(-9 + 1\). Perform the arithmetic operation: \(-9 + 1 = -8\). So, the simplified right side is \(-8\).

Step 3: Rewrite the Simplified Equation

Based on your simplifications, rewrite the equation with the new components: \(a - 2 = -8\).

Step 4: Apply the Addition Property

Add 2 to both sides of the equation to isolate \(a\).: \(a - 2 + 2 = -8 + 2\). The left side becomes \(a\), and the right side becomes \(-6\), resulting in the equation \(a = -6\).

Key Concepts

Addition Property

In mathematics, the addition property of equality is a fundamental concept. It states that if you add the same amount to both sides of an equation, the equation remains balanced. This is an important principle because it helps us manipulate and solve equations while maintaining equality.

Imagine a seesaw that stays balanced if you place equal weights on both sides. Similarly, in equations, adding the same value to the left and right sides keeps them equal, which is crucial for solving problems.

• Add the same number to both sides to maintain equality.
• Helps to isolate variables in equations.
• Used to transition from a complex equation to a simpler solution.

In the exercise, we added 2 to both sides as part of the solution process. This use of the addition property allowed us to isolate the variable \(a\) and find its value, showing how powerful this tool can be when solving equations.

Combining Like Terms

Combining like terms is another important skill when simplifying equations. Like terms have the same variable raised to the same power, and only their coefficients differ. By combining them, we simplify the expression, making it easier to solve.

For example, in the original exercise, we had terms \(6a\) and \(-5a\) on the left side of the equation. Since both terms had the same variable \(a\), they could be combined easily:

• Add or subtract coefficients of like terms to simplify an expression.
• Reduces the complexity of equations.
• Makes it easier to see relationships between terms.

The combination process was: \(6a - 5a = 1a\). The exercise then simplified to \(a - 2\) on the left side. This allowed for a clearer path to solve for \(a\) once the terms were combined, highlighting the importance of this concept.

Prealgebra

Prealgebra is often the first step into the world of abstract mathematics and algebraic thinking. It focuses on using arithmetic and variables to solve problems, setting the foundation for more advanced topics.

In prealgebra, students begin to understand how numbers and operations interact through equations, like those in the exercise. It’s about learning to manipulate expressions and equations step by step:

• Focus on foundational concepts like variables, equations, and arithmetic operations.
• Prepares students for algebra by establishing basic problem-solving skills.
• Encourages logical thinking and methodical approach to math problems.

The original exercise required skills that are fundamental in prealgebra, such as simplifying expressions and solving for variables, demonstrating how these principles build the groundwork for future mathematical learning.