Problem 8 - Prealgebra

Question

Solve each equation. Check your solution. $$2 a-8 a=24$$

Step-by-Step Solution

Answer

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

Step 1: Combine Like Terms

We start with the equation \(2a - 8a = 24\). Combine the like terms on the left side. The terms \(2a\) and \(-8a\) can be combined as they both contain the variable \(a\). This results in \(-6a = 24\).

Step 2: Isolate the Variable

To find the value of \(a\), we need to isolate it on one side of the equation. We currently have \(-6a = 24\). Divide both sides of the equation by \(-6\) to get \(a\) by itself. This results in \(a = -4\).

Step 3: Check the Solution

We verify our solution by substituting \(a = -4\) back into the original equation to ensure both sides equal. Substitute into the original equation: \(2(-4) - 8(-4) = 24\), which simplifies to \(-8 + 32 = 24\) and confirms that \(24 = 24\), so our solution is correct.

Key Concepts

Combining Like Terms

When solving equations, the first step often involves combining like terms. Like terms are terms in an equation that contain the same variables raised to the same power. For instance, in the equation \(2a - 8a = 24\), the terms \(2a\) and \(-8a\) are like terms because they both include the variable \(a\).

• To combine these terms, simply add or subtract their coefficients. In our example, \(2a - 8a\) combines to \(-6a\).
• Think of the coefficients as numbers of apples; if you have 2 apples and take away 8 apples, you end up with \(-6\) apples.

This simplification is an essential step because it reduces the complexity of the equation
and brings us closer to solving for the variable while ensuring the equation remains balanced.

Isolating Variables

After combining like terms, the next task is to isolate the variable. Isolating a variable means getting it by itself on one side of the equation. In our example, after combining to obtain \(-6a = 24\), the goal is to have \(a\) by itself.

• To isolate \(a\), divide both sides by \(-6\), which is the coefficient of \(a\).
• This operation will cancel out the \(-6\) on the side with \(a\), resulting in \(a = -4\).
• Remember that whatever operation you apply to one side of the equation, you must also apply to the other side to maintain equality.

This principle is key because it relies on the basic properties of equality and ensures that the solution is valid. Isolating the variable transforms the equation into a much simpler form, allowing us to directly see the value that the variable represents.

Checking Solutions

Once you've found a solution by isolating the variable, it's crucial to check your work. This step verifies that the solution satisfies the original equation. Checking your solutions involves substituting your answer back into the original equation and confirming both sides are equal.

• For \(a = -4\), substitute it into the original formula \(2a - 8a = 24\).
• This becomes \(2(-4) - 8(-4) = 24\). Simplifying both terms gives \(-8 + 32 = 24\).
• Since \(-8 + 32\) indeed equals 24, our solution of \(a = -4\) checks out.

Whenever checking a solution, make sure the operations align perfectly with those of the original equation.
This step serves as a safeguard against errors made during the calculation process. It confirms that the entire solving process, from combining terms to isolating the variable, was executed correctly.