Problem 4 - Elementary Algebra

Question

Simplify. $$ 4 x-3+6 x-3 $$

Step-by-Step Solution

Answer

The simplified expression is \(10x - 6\).

Step 1: Identify Like Terms

To simplify an expression, first identify like terms. In the expression \(4x - 3 + 6x - 3\), the terms \(4x\) and \(6x\) are like terms because they both contain \(x\), and the terms \(-3\) and \(-3\) are also like terms as they are constants.

Step 2: Combine Like Terms Containing the Variable

Add the coefficients of the like terms containing \(x\): \(4x + 6x = 10x\). This simplifies the variable portion of the expression to \(10x\).

Step 3: Combine Constant Terms

Add the constant terms: \(-3 + (-3) = -6\). This simplifies the constant portion of the expression to \(-6\).

Step 4: Write the Simplified Expression

Combine the results from Step 2 and Step 3 to get the simplified version of the expression: \(10x - 6\).

Key Concepts

Like Terms

When working on algebraic expressions, identifying like terms is the first step toward simplification. Like terms are terms that have identical variable parts. For example, in the expression \(4x - 3 + 6x - 3\), the terms \(4x\) and \(6x\) are like terms because they both involve the variable \(x\). In contrast, numbers without variables, also known as constant terms like \(-3\) and another \(-3\), are also considered like terms since no variables are involved. Recognizing which terms in an expression are like terms helps simplify the process of combining them into one term.

• Identify if terms share the same variable and exponent.
• Check for similarity among constants as they are naturally like terms.

Combining Like Terms

After identifying like terms, the next step is to combine them so we can simplify the expression. Combining like terms means performing arithmetic operations on these terms, such as addition or subtraction. For the expression \(4x - 3 + 6x - 3\), we first combine the variable terms: \(4x\) and \(6x\). The operation looks like this:

• Add the coefficients: \(4 + 6 = 10\).
• The result is \(10x\).

Then, we combine the constant terms: \(-3\) and \(-3\). By adding these:

• \(-3 + (-3) = -6\).

The expression is then reduced to its simplest form: \(10x - 6\). Breaking this down helps you see how each part of the expression is dealt with one step at a time.

Coefficients

Coefficients are the numerical factors of the terms in algebraic expressions. They are the numbers that multiply the variables. For example, in the term \(4x\), the coefficient is \(4\). This means that the variable \(x\) is multiplied by \(4\). In the expression \(4x + 6x\), \(4\) and \(6\) are coefficients.Co-efficients have an essential role in combining like terms since they are the numbers we're mainly summing up when combining these terms. Consider:

• If you have more of a variable (like \(x\)), the coefficients tell you exactly how many you have.
• Combining \(4x\) and \(6x\) means adding their coefficients: \(4 + 6\).
• This operation simplifies to \(10x\), which is based entirely on the sum of the coefficients.

Constants

Constants in algebra are numbers on their own, without any variable attached. They represent fixed values. In the expression you're simplifying, \(-3\) is a constant term. When simplifying expressions, constants are added or subtracted in the same way numbers are in basic arithmetic. For the expression \(-3 + (-3)\):

• The numbers are combined normally, just as you would with simple addition.
• Here, \(-3 + (-3) = -6\).

In algebraic expressions, constants can make up part of like terms when they are the same number or represent similar values. Simplifying constants might seem straightforward, but they have an essential role in finding the simplest form of an expression.