Problem 34 - Prealgebra

Question

Simplify each side of the following equations before applying the addition property. $$5 a+6-4 a=4$$

Step-by-Step Solution

Answer

The solution is \(a = -2\).

Step 1: Combine Like Terms

Look at the left-hand side of the equation: \(5a + 6 - 4a = 4\). We combine the like terms, which are the terms with \(a\). \(5a - 4a\) simplifies to \(1a\) or simply \(a\). After combining, the equation becomes \(a + 6 = 4\).

Step 2: Isolate the Variable

Now that we have simplified the left-hand side, we need to isolate \(a\) by using the addition property of equality. Subtract 6 from both sides: \((a + 6) - 6 = 4 - 6\). This simplifies to \(a = -2\).

Key Concepts

Combining Like Terms

When solving linear equations, combining like terms is a fundamental step. Like terms are terms that have the same variable raised to the same power. In the equation given, like terms include those with the variable 'a', such as \(5a\) and \(-4a\). It is important to identify and group these terms together before moving forward.

Once identified, add or subtract the coefficients of these like terms. For instance, \(5a\) and \(-4a\) are like terms. Their coefficients are 5 and -4, respectively. We combine them by performing the arithmetic operation:

• \(5 - 4 = 1\)

This results in '1a', which simplifies further to just 'a'. By doing this, we have reduced the equation and made it simpler to solve. Always ensure that you only combine terms that are alike, meaning they must have identical variables and exponents.

Addition Property of Equality

The addition property of equality is a useful tool in maintaining the balance of an equation. This principle states that if you add or subtract the same number from both sides of an equation, the two sides remain equal.

After simplifying the equation for like terms, we are left with \(a + 6 = 4\). To isolate 'a', we need to remove the 6 that is added to it. We can do this by subtracting 6 from both sides of the equation, applying the addition property of equality:

• Subtract 6 from the left: \((a + 6) - 6 = a\)
• Subtract 6 from the right: \(4 - 6 = -2\)

By doing this, you aren't changing the equation's balance. You are instead preserving equality and bringing 'a' alone on one side of the equation. Thus, the equation simplifies to \(a = -2\). This step is essential to solve for the variable.

Simplifying Expressions

Simplifying expressions is an important part of solving equations efficiently. An expression is simplified when it is reduced to its simplest form, making it more straightforward to work with. This involves eliminating unnecessary complexities, such as removing brackets, reducing coefficients, and combining like terms.

In our given problem, after identifying and combining like terms, we simplify the left-hand expression from \(5a + 6 - 4a\) to \(a + 6\). This is a significant simplification since it reduces the complexity of the equation.

To simplify further, we use the addition property of equality as discussed. Each of these steps not only reduces the equation but also brings you closer to the solution. By consistently simplifying expressions, you make complex problems more manageable and easier to solve. Always aim for the most basic, clear form of an equation, as this will aid in understanding and solving it more effectively.