Problem 48 - Algebra for College Students

Question

Perform the indicated operations. $$(-3 x+1)+(6 x-2)+(9 x-4)$$

Step-by-Step Solution

Answer

The simplified expression is \(12x - 5\).

Step 1: Identify Like Terms

To solve the expression \((-3x + 1) + (6x - 2) + (9x - 4)\), we need to identify and group like terms. Here, the like terms are those containing \(x\) and the constant terms. Collect them from each part of the expression.

Step 2: Group Like Terms

Group all the terms with \(x\) together and all the constant terms together: \((-3x + 6x + 9x) + (1 - 2 - 4)\).

Step 3: Simplify the x Terms

Add the coefficients of the \(x\) terms: \((-3 + 6 + 9)x = 12x\).

Step 4: Simplify the Constant Terms

Add the constant terms: \(1 - 2 - 4 = -5\).

Step 5: Combine Simplified Terms

Combine the results from Steps 3 and 4 to form the final simplified expression: \(12x - 5\).

Key Concepts

Like Terms

Think of like terms as similar items that you can group together. In algebraic expressions, like terms contain the same variable raised to the same power. For instance, in the expression \[(-3x + 1) + (6x - 2) + (9x - 4)\]all terms containing the variable \(x\) are like terms. These are

• -3x
• 6x
• 9x

Like terms make simplifying algebraic expressions easier because you can combine them directly. This means you simply add or subtract their coefficients. It’s like counting how many "\(x\)" you have in total!

Coefficients

A coefficient in an algebraic expression is the numerical factor that multiplies a variable. In our exercise, each term with \(x\) has a number in front of it—this is the coefficient. For example:

• The term \(-3x\) has a coefficient of -3
• In \(6x\), the coefficient is 6
• For \(9x\), the coefficient is 9

When simplifying, focus on the coefficients to adjust the values of like terms. It's important to pay attention to the sign, whether positive or negative, as it affects the result when you combine terms within the expression.

Constant Terms

Constant terms are numbers without attached variables in an algebraic expression—essentially, plain numbers. In the given expression, examine the terms:

• +1
• -2
• -4

These are the constant terms. Combining constant terms involves regular arithmetic operations like addition and subtraction. Keep track of the signs, as they alter the sum or difference when you simplify. Constants are like fixed points that help balance the variable parts of an expression.

Simplifying Expressions

Simplifying an expression means combining like terms and reducing the expression to its simplest form. Here, we simplify by performing the indicated operations: addition or subtraction of like terms. Start by regrouping like terms to get everyone ready for calculation. You get:

• For \(x\) terms: \(-3x + 6x + 9x\)
• For constant terms: \(1 - 2 - 4\)

Next, simplify each group:

• Add the coefficients of \(x\) to find the total: \((-3 + 6 + 9)x = 12x\)
• Add the constants: \(1 - 2 - 4 = -5\)

Finally, put it all together, and you have the simplified expression: \(12x - 5\). This is now easier to understand and can be used for further calculations if needed.