Problem 121 - Elementary Algebra

Question

Simplify: \(8 x-9 x+12 x-15 x\)

Step-by-Step Solution

Answer

-4x

Step 1: Combine Like Terms

Identify the like terms in the expression. In this case, all the terms (8x, -9x, 12x, -15x) are like terms because they all have the variable x.

Step 2: Add and Subtract the Coefficients

Add and subtract the coefficients of the like terms: 8 - 9 + 12 - 15.

Step 3: Simplify the Coefficients

Solve the arithmetic: 8 - 9 = -1, -1 + 12 = 11, 11 - 15 = -4.

Step 4: Rewrite the Expression

Replace the original expression with the simplified coefficient followed by the variable: -4x.

Key Concepts

Combining Like Terms

When simplifying algebraic expressions, the first crucial step is combining like terms. Like terms are terms that contain the same variables raised to the same power. In other words, they only differ by their coefficients. For example, in the expression,

• 8x
• -9x
• 12x
• -15x

all terms have the variable x. Because they share the same variable, we can combine them by adding or subtracting their coefficients.

This concept is vital because it allows us to simplify complex expressions into more manageable ones. By combining these terms, we replace multiple terms with a single term, making the overall algebraic expression simpler and easier to work with.

Coefficients

A coefficient is the numerical part of a term that contains a variable. In the expression 8x, the number 8 is the coefficient, and in -9x, -9 is the coefficient.

When combining like terms, we focus solely on the coefficients. The variable part remains unchanged. For instance, if we have the expression 8x - 9x + 12x - 15x, we look at the coefficients 8, -9, 12, and -15 individually.

Next, we perform the arithmetic operations (addition or subtraction) on these coefficients. This is done step-by-step:

• 8 - 9 = -1


• -1 + 12 = 11


• 11 - 15 = -4

By solving these, we get -4 as the new coefficient.

The simplified coefficient then combines with the variable to form the simplified expression, which in this case, is -4x.

Simplification

Simplification in algebra refers to the process of reducing an expression to its simplest form. This involves combining like terms and performing appropriate arithmetic operations on the coefficients.

In our example, we start with the expression 8x - 9x + 12x - 15x. First, we identify the like terms, then add or subtract their coefficients. Finally, we rewrite the expression with the simplified coefficient and the common variable.

Simplifying expressions is a fundamental skill in algebra because:

• It makes complex equations more manageable.


• It helps in solving equations more efficiently.


• It allows better understanding and interpretation of mathematical relationships.

The end goal is to make the expression as straightforward as possible. Practicing simplification with various expressions helps in mastering this essential algebra skill. In our example, the fully simplified expression is -4x.