Problem 65 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ -6 x+2 y-8 x+4 y $$

Step-by-Step Solution

Answer

The simplified form of the expression \( -6x + 2y - 8x + 4y \) is \( -14x + 6y \).

Step 1: Identify Like Terms

The given expression is \( -6x + 2y - 8x + 4y \). By examining this, it can be seen that \( -6x \) and \( -8x \) are like terms, since they both contain the variable \( x \). Similarly, \( 2y \) and \( 4y \) are like terms because they both contain the variable \( y \).

Step 2: Add or Subtract the Coefficients of Like Terms

The coefficient of \( x \) in the Terms \( -6x \) and \( -8x \) are \( -6 \) and \( -8 \) respectively. Adding these, we get \( -6 - 8 = -14 \). So, the combined term for \( x \) is \( -14x \). \n Similarly, the coefficient of \( y \) in the Terms \( 2y \) and \( 4y \) are \( 2 \) and \( 4 \) respectively. Adding these, we get \( 2 + 4 = 6 \). So, the combined term for \( y \) is \( 6y \).

Step 3: Write the Simplified Expression

The simplified expression is the sum of the combined \( x \) term and the combined \( y \) term. Plugging the values we found in Step 2, we get \( -14x + 6y \). This is our final result.

Key Concepts

Understanding Like Terms

When working with algebraic expressions, it's crucial to recognize and identify 'like terms'. Like terms are terms that have the exact same variable part, although their coefficients (the numerical part) might differ. Here's a simple way to spot them:

• Terms are considered "like" if they have the same variables raised to the same powers. For example, in the expression \(-6x + 2y - 8x + 4y\), both \(-6x\) and \(-8x\) contain the variable \(x\). Similarly, \(2y\) and \(4y\) both contain the variable \(y\).


• The coefficients, however, can be different. What matters is the variable portion.

Identifying like terms is the first step toward simplifying expressions. Once you've found them, you can move on to the next step: combining them.

Combining Like Terms

Combining like terms is the process of simplifying an expression by adding or subtracting the coefficients of terms that are alike. Once you’ve identified the like terms in an expression, you can combine them:

• Take each group of like terms separately. Add or subtract their coefficients to get one single term. In our exercise, we have \(-6x\) and \(-8x\). Their coefficients, \(-6\) and \(-8\), when combined give \(-14\).


• Similarly, for the terms \(2y\) and \(4y\), the coefficients are \(2\) and \(4\). Added together, they give \(6\). Therefore, \(6y\) is our combined term for these like terms.

This process reduces the complexity of an expression and makes it easier to interpret and solve.

Understanding Coefficients

The coefficient in an algebraic term is the constant number that multiplies the variable. In the expression \(-6x\), the number \(-6\) is the coefficient. It's important to handle coefficients properly during simplification:

• They tell us how many times the corresponding variable is included in the term. For instance, \(-6x\) essentially means \(x\) is being added to itself \(-6\) times.


• When combining like terms, we perform arithmetic operations (either addition or subtraction) on the coefficients to simplify the expression. For \(-6x\) and \(-8x\), subtracting \(8\) more \(x's\) gives us \(-14x\).

Understanding and manipulating coefficients accurately is vital when simplifying algebraic expressions. It ensures that the simplified version of the expression is equivalent to the original.