Problem 84 - Intermediate Algebra

Question

Solve each equation. $$ 4 a-21-a=-2 a-7 $$

Step-by-Step Solution

Answer

The solution is \(a = 2.8\).

Step 1: Combine Like Terms on the Left Side

First, we need to combine the like terms on the left side of the equation. The terms \(4a\) and \(-a\) both contain \(a\), so we can add them together to get \(3a\).\[4a - 21 - a = 3a - 21\]

Step 2: Rearrange the Equation

Now, let's rearrange the equation so that all terms containing \(a\) are on one side and constant terms are on the other. We do this by adding \(2a\) to both sides:\[3a - 21 + 2a = -7 + 2a\]This simplifies to:\[5a - 21 = -7\]

Step 3: Isolate the Variable Term

Next, we need to isolate the term \(5a\). Do this by adding 21 to both sides of the equation:\[5a - 21 + 21 = -7 + 21\]This gives us:\[5a = 14\]

Step 4: Solve for \(a\)

Finally, solve for \(a\) by dividing both sides of the equation by 5:\[a = \frac{14}{5}\]Therefore, \(a = 2.8\).

Key Concepts

Combining Like Terms

When solving linear equations, the first thing you often need to do is simplify the expressions on each side of the equation by combining like terms. Like terms are terms that have the same variables raised to the same power. For example, in the equation \(4a - 21 - a\), the terms \(4a\) and \(-a\) are like terms because both include the variable \(a\).

• To combine them, simply perform the arithmetic operation between their coefficients. Here, you subtract the coefficient of \(-a\) (which is \(-1\)) from the coefficient of \(4a\) (which is \(4\)), resulting in \(3a\).

This reduction transforms the equation, making it simpler and easier to handle. Combining like terms is a crucial step that tidies up your equation before moving on to other operations. It helps in reducing complexity and targets the variable terms directly, allowing more focus on isolating the variable.

Isolating the Variable

Once like terms are combined, the next objective is to isolate the variable on one side of the equation. This step typically involves removing any non-variable terms from the side where the variable is, or conversely, moving the variables to one side and constants to the other.

• In the example, we first simplify to \(3a - 21 = -2a - 7\). By adding \(2a\) to both sides, we keep the balance and collect all variable terms on one side:
  \[3a + 2a - 21 = -7\]. This simplifies further to \(5a - 21 = -7\).
• The next move is to deal with the constant \(-21\) next to the variable term. Adding 21 to both sides isolates the variable term, resulting in \(5a = 14\).

By focusing on getting the variable term by itself, you are effectively setting the stage to find the variable's value. This process of isolating the variable ensures you're prepared to solve for it completely.

Rearranging Equations

Rearranging equations is a process that involves moving terms around while maintaining the balance of the equation. This is essential because the goal is to have the equation in a form where the variable is easily solvable.

• Initially, you have the equation \(3a - 21 = -2a - 7\). Rearranging involves making decisions about what moves to make, like adding or subtracting terms on both sides. In this particular problem, moving \(2a\) across the equal sign by adding it to \(3a\) helped in consolidating the variable terms.
• Similarly, obtaining \(5a = 14\) after adding 21 to both sides set the stage for straightforward division.

Understanding rearrangement is akin to efficiently solving a puzzle, where each move you make helps inch you closer to visualizing the solution. It ensures the equation remains accurate and robust throughout the solving process.