Problem 84
Question
Simplify each algebraic expression. $$-6 x-9 y-4 x+15 y$$
Step-by-Step Solution
Verified Answer
The simplified form of the given algebraic expression is -10x + 6y.
1Step 1: Identify Like Terms
The given expression is \( -6x - 9y - 4x + 15y \). Here, '-6x' and '-4x' are alike because they both contain the variable 'x'. Similarly, '-9y' and '15y' are alike because they both contain the variable 'y'.
2Step 2: Combine Like Terms
Now, add or subtract the coefficients (numbers in front of variables) of like terms. The sum of '-6x' and '-4x' is '-10x'. The sum of '-9y' and '15y' is '6y'. So the simplified expression becomes '-10x + 6y'.
Key Concepts
Combining Like TermsAlgebraic ExpressionCoefficients
Combining Like Terms
Understanding how to combine like terms is crucial when simplifying algebraic expressions. 'Like terms' are terms in an equation that have the exact same variable raised to the same power. In other words, they are terms that are 'alike' in their variable parts. For instance, consider the expression \( -6x - 9y - 4x + 15y \). The terms '−6x' and '−4x' are like terms because they both contain the letter 'x', while '−9y' and '15y' are like terms because they both contain the letter 'y'. To combine like terms, you'll simply perform basic arithmetic—addition or subtraction—with the coefficients of these terms.
When the coefficients are positive and negative, like '-6' for \( -6x \) and '-4' for \( -4x \), adding them gives you \( -10x \). Similarly, for 'y', subtracting the smaller coefficient from the larger one, you get \( -9y + 15y \), which simplifies to \( 6y \). The real beauty of combining like terms comes from its ability to simplify complex expressions, allowing equations to be solved more easily and making the overall arithmetic less daunting.
When the coefficients are positive and negative, like '-6' for \( -6x \) and '-4' for \( -4x \), adding them gives you \( -10x \). Similarly, for 'y', subtracting the smaller coefficient from the larger one, you get \( -9y + 15y \), which simplifies to \( 6y \). The real beauty of combining like terms comes from its ability to simplify complex expressions, allowing equations to be solved more easily and making the overall arithmetic less daunting.
Algebraic Expression
At its heart, an algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. Variables, like 'x' and 'y', represent unknown values and can change, like characters in a play. They are fundamental in expressing relationships and changes in quantities.
Take for example the expression \( -6x - 9y - 4x + 15y \). This expression tells a story. It has two variables, 'x' and 'y', and showcases how they interact with numbers (coefficients) through subtraction and addition. Simplifying this expression is like translating a complex sentence into a simpler form or condensing a paragraph into a single, clear line. The process of simplification may involve combining like terms or using other algebraic properties to ensure that similar variables stick together, making the expression cleaner and clearer.
Take for example the expression \( -6x - 9y - 4x + 15y \). This expression tells a story. It has two variables, 'x' and 'y', and showcases how they interact with numbers (coefficients) through subtraction and addition. Simplifying this expression is like translating a complex sentence into a simpler form or condensing a paragraph into a single, clear line. The process of simplification may involve combining like terms or using other algebraic properties to ensure that similar variables stick together, making the expression cleaner and clearer.
Coefficients
In the context of algebra, coefficients are the numbers situated before variables and serve as multipliers to those variables. Think of coefficients as the numerical strength or magnitude assigned to a variable in an expression. In the example \( -6x - 9y - 4x + 15y \), '-6' is the coefficient of 'x' in the term \( -6x \), and '-9' is the coefficient of 'y' in \( -9y \).
During the process of combining like terms, the coefficients are what you actually add or subtract to simplify the expression. It is similar to saying, 'if I have 6 apples and then lose 4 apples, I now have 2 apples.' Here the apples are like the variable, and the numbers are like the coefficients. When simplifying, it's important to remember that only the coefficients interact with each other—the variables ride along, unchanged, unless the entire term is eliminated through addition or subtraction of the coefficients.
During the process of combining like terms, the coefficients are what you actually add or subtract to simplify the expression. It is similar to saying, 'if I have 6 apples and then lose 4 apples, I now have 2 apples.' Here the apples are like the variable, and the numbers are like the coefficients. When simplifying, it's important to remember that only the coefficients interact with each other—the variables ride along, unchanged, unless the entire term is eliminated through addition or subtraction of the coefficients.
Other exercises in this chapter
Problem 83
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In Exercises \(81-88,\) simplify each algebraic expression by removing parentheses and brackets. $$5[2-(y+3)]$$
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Simplify each algebraic expression. $$-6 x+x$$
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