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Chapter 2: Problem 91

Simplify by combining like terms whenever possible. $$2 x+5 x-3 y+y-7 x$$

Short Answer

Expert verified
-2y

Step by step solution

01

- Identify like terms

Group the terms that have the same variable. In this case, you can group the terms with 'x' and the terms with 'y'. For example:Terms with 'x': 2x, 5x, -7xTerms with 'y': -3y, y
02

- Combine like terms with 'x'

Add the coefficients of the terms with 'x': 2x + 5x - 7xThis simplifies to:(2 + 5 - 7)x = 0x
03

- Combine like terms with 'y'

Add the coefficients of the terms with 'y': -3y + yThis simplifies to:(-3 + 1)y = -2y
04

- Write the simplified expression

Combine the results from Steps 2 and 3:0x - 2ySince 0x adds nothing to the expression, the simplified form is:-2y

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combining Like Terms
Combining like terms is a fundamental concept in algebra that makes expressions simpler and easier to manage. When combining like terms, look for terms that have the same variable and are raised to the same power. For example: In the exercise, the like terms are those with the variable 'x' (2x, 5x, -7x) and those with 'y' (-3y, y). To combine them, add or subtract their coefficients.
Combining like terms is important because:
  • It simplifies the expression.
  • It helps in solving equations easier.
  • It makes the expression clearer and more concise.
Remember, only terms with the same variables can be combined. Terms with different variables or different powers cannot be combined.
Coefficients
Coefficients are the numerical part of terms that contain variables. In the expression 2x, 2 is the coefficient, and in -3y, -3 is the coefficient. The coefficient tells us how many times the variable is being multiplied.
Key points to remember about coefficients:
  • They can be positive or negative.
  • You can add or subtract coefficients if the terms have the same variable.
  • They simplify arithmetic with algebraic expressions.
For example: In the problem, the coefficients for the 'x' terms (2x, 5x, -7x) are 2, 5, and -7. We add these coefficients together: 2 + 5 - 7 = 0. Similarly, the coefficients for the 'y' terms (-3y, y) are -3 and 1. Adding these gives: -3 + 1 = -2.
Algebraic Simplification
Algebraic simplification involves reducing expressions to their simplest form. This process often entails combining like terms and performing basic arithmetic.
Steps to Simplify Algebraic Expressions:
  • Identify like terms.
  • Combine the coefficients of like terms.
  • Rewrite the expression with the simplified terms.
Consider the original problem: $$ 2x + 5x - 3y + y - 7x $$. We group and combine the 'x' terms: $$ (2 + 5 - 7)x = 0x $$. Since anything multiplied by zero is zero, we eliminate '0x' from the expression. Next, we combine the 'y' terms: $$ -3y + y = -2y $$. So, the simplified expression is: $$ -2y $$. Note that a term with a coefficient of 0 is always removed from the final simplified expression, as it does not contribute to the value.

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