Problem 30 - College Algebra

Question

For the following exercises, solve for the variable. $$ (11 a+3)-18 a=-4 $$

Step-by-Step Solution

Answer

The solution for the variable is \(a = 1\).

Step 1: Simplify Both Sides

We begin by simplifying the expression on the left side of the equation. Combine the terms with the variable "a":\[(11a + 3) - 18a\]This simplifies by combining like terms:\[11a - 18a + 3 = -7a + 3\]Now the equation becomes:\[-7a + 3 = -4\]

Step 2: Isolate the Variable Term

We aim to isolate the term with the variable by eliminating the constant term on the left side of the equation. Subtract 3 from both sides:\[-7a + 3 - 3 = -4 - 3\]This simplifies to:\[-7a = -7\]

Step 3: Solve for the Variable

To solve for the variable "a", divide both sides by -7:\[a = \frac{-7}{-7}\]Simplifying gives:\[a = 1\]

Key Concepts

Isolate the variable

When solving linear equations, understanding how to isolate the variable is crucial. Isolating the variable means rearranging the equation so that the unknown variable stands alone on one side of the equation. This allows us to easily see what the variable equals.

In order to isolate the variable, you often need to perform several operations. These operations include:

• Adding or subtracting numbers or expressions from both sides of the equation.
• Multiplying or dividing both sides of the equation by a non-zero number.

By isolating the variable, we focus on simplifying until only 'a' is left on one side. For example, if you have \[-7a + 3 = -4\]you can remove the constant 3 by subtracting it from both sides. Doing this gives:\[-7a = -7\]This process leaves the variable 'a' ready for the next steps, making it simpler to find its value.

Simplifying algebraic expressions

Simplifying algebraic expressions means to make them as simple as possible. This usually involves performing basic arithmetic operations such as addition, subtraction, multiplication, or division. By reducing expressions, we make them easier to work with and understand.

In our example:\[(11a + 3) - 18a\]To simplify, we combined the terms that contain the variable 'a'. Here, that means adding \[11a\] and subtracting \[18a\] to get:\[-7a + 3\]This simplification is a foundational step that frequently helps in revealing parts of the equation that are often hidden or obscured. Once simplified, the expression \[-7a + 3\] can be more easily manipulated. Simplification is about making algebra easier to manipulate and understand so that we can proceed with solving the equation.

Combining like terms

Combining like terms is one of the first steps in simplifying any algebraic expression. It involves identifying and merging terms that are similar—those that contain the same variable raised to the same power. This process helps reduce the complexity of expressions and is a core concept in simplifying equations.

For instance, in the expression \[(11a + 3) - 18a\], both \[11a\] and \[-18a\] are like terms because they both contain the variable 'a'. When we combine them, we perform the operation \[11a - 18a\] to yield \[-7a\].Similar terms make handling equations more straightforward as they reduce multiple terms to a single term. After combining them, it's less cluttered and easier to see how to proceed with solving the equation. It's akin to tidying up the terms so you can clearly see what needs to be done next.