Problem 36 - Prealgebra

Question

Simplify each side of the following equations before applying the addition property. $$7 a-6-6 a=-3+1$$

Step-by-Step Solution

Answer

The solution to the equation is \(a = 4\).

Step 1: Simplify Both Sides

First, we simplify both sides of the equation. Start with the left side, which is \(7a - 6 - 6a\). We combine like terms by subtracting \(6a\) from \(7a\), resulting in \(a - 6\). The equation now becomes: \(a - 6 = -3 + 1\).

Step 2: Simplify the Right Side

Next, simplify the right side of the equation, which is \(-3 + 1\), adding these numbers gives us \(-2\). The equation now reads \(a - 6 = -2\).

Step 3: Apply the Addition Property

To solve for \(a\), we use the addition property of equality to isolate \(a\). Add 6 to both sides of the equation to eliminate the \(-6\) on the left side. This gives us \(a - 6 + 6 = -2 + 6\), simplifying to \(a = 4\).

Key Concepts

Addition Property

The addition property of equality is a fundamental principle used to solve equations. It states that if you add the same number to both sides of an equation, the equality remains unchanged. This property helps to simplify equations and find the value of unknown variables.

Let's think of it like a balanced scale. Imagine you have a balance scale with equal weights on both sides. If you add or remove the same amount from each side, the scale remains balanced. Similarly, in an equation, when you add or subtract the same value on both sides, the equation remains true.

In our exercise, we used this property in Step 3. We had the equation, \(a - 6 = -2\). To isolate \(a\), we added 6 to both sides, like this: \(a - 6 + 6 = -2 + 6\). This simplified to \(a = 4\), giving us the solution. Next time you encounter an equation, remember that you can freely add or subtract the same number to both sides—this powerful tool keeps the equation balanced and helps you solve for the variable.

Combining Like Terms

Combining like terms is a method used to simplify equations or expressions by merging terms that contain the same variable raised to the same power. This step makes mathematical problems easier to handle, especially when dealing with complex equations.

For instance, in Step 1 of our example, we simplified the left side of the equation, \(7a - 6 - 6a\), by combining like terms. Both \(7a\) and \(-6a\) belong to the same variable \(a\). By subtracting \(6a\) from \(7a\), we simplified it to \(a\). Thus, the left side of the equation became \(a - 6\).

Here are a few helpful tips when combining like terms:

• Identify terms with the same variable and the same exponent.
• Add or subtract the coefficients of these terms.
• Maintain the variable and its exponent in the final term.

By mastering the art of combining like terms, you can reduce complicated expressions into manageable parts, making it easier to solve the rest of the equation.

Prealgebra

Prealgebra serves as an introduction to the basic concepts and skills needed for algebra. It lays the foundation for solving equations and understanding more advanced math topics. Prealgebra involves operations with numbers, basic geometry, understanding integers, fractions, and beginning to work with variables and simple equations.

In our example, the equation given is a perfect exercise for practicing prealgebra skills. You get to work with variables, simplify expressions, and use properties such as the addition property of equality to solve equations. Each of these steps is a crucial part of developing the mathematical reasoning prepared in prealgebra.

Transitioning into algebra is easier with a solid grasp of prealgebra concepts. You build the confidence to approach problems methodically. Remember that making mistakes and learning from them is part of the learning process. By practicing exercises like the one we've solved, your skills in working with variables and simplifying equations will grow, preparing you for more complex math tasks in the future.