Problem 23 - Algebra A Combined Function

Question

Simplify each expression by combining any like terms. $$ 6 x+0.5-4.3 x-0.4 x+3 $$

Step-by-Step Solution

Answer

The simplified expression is \(1.3x + 3.5\).

Step 1: Identify Like Terms

Like terms are those terms that have the same variable raised to the same power. In this expression, the like terms are those that have the variable \(x\) and the constants. The terms involving \(x\) are \(6x\), \(-4.3x\), and \(-0.4x\). The constants are \(0.5\) and \(3\).

Step 2: Combine the Variable Terms

Add the coefficients of the like terms that include the variable \(x\): \[ 6x - 4.3x - 0.4x = (6 - 4.3 - 0.4)x \] Calculate the sum: \[ 6 - 4.3 - 0.4 = 1.3 \] So the combined term for \(x\) is \(1.3x\).

Step 3: Combine the Constant Terms

Now, add the constant terms together: \[ 0.5 + 3 = 3.5 \] So the constant term is \(3.5\).

Step 4: Write the Simplified Expression

Put together the simplified variable term and the simplified constant term. The expression becomes:\[ 1.3x + 3.5 \]

Key Concepts

Like Terms

Understanding like terms is essential when dealing with algebraic expressions. In algebra, like terms refer to terms that have the same variable and are raised to the same power. For example, in the expression \(6x + 0.5 - 4.3x - 0.4x + 3\), the terms \(6x\), \(-4.3x\), and \(-0.4x\) are like terms because they all contain the variable \(x\) to the same power. The constants \(0.5\) and \(3\) are also considered like terms, but they fall under constants as they lack the variable \(x\).

When simplifying expressions, it's crucial to first identify these like terms so you can combine them later. A handy tip is to look for terms that look the same except for their coefficients. Like terms can also include more complex expressions such as \(2xy\) and \(5xy\), as long as the variables and their powers match.

Combining Coefficients

Combining coefficients is the next step after identifying like terms. Coefficients are the numerical part of terms with variables. In our expression, the coefficients for \(x\) are \(6\), \(-4.3\), and \(-0.4\).

You add or subtract these coefficients depending on their signs. In practice, this looks like:

• Add 6 (positive)
• Subtract 4.3 (negative)
• Subtract 0.4 (negative)

So, you calculate \(6 - 4.3 - 0.4 = 1.3\). This tells us that the combined coefficient for \(x\) is \(1.3\).

Remember, combining coefficients only applies to like terms. For non-variable terms or constants, you simply perform regular addition or subtraction.

Simplification Process

The simplification process in algebra involves reducing an expression to its simplest form. Start by combining like terms, as we've discussed. Take our example \(6x + 0.5 - 4.3x - 0.4x + 3\).

First, group the like terms and add their coefficients. As noted, \(6x - 4.3x - 0.4x\) combines into \(1.3x\).

Next, address the constant terms, \(0.5\) and \(3\). Combine these just like regular numbers:

• Calculate \(0.5 + 3 = 3.5\)

Finally, combine the simplified parts into the expression: \(1.3x + 3.5\). This final step of the simplification process ensures your expression is as concise as possible, making it easier to work with or solve further.