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Chapter 13: Problem 12

Find sum. \((6 y-5 r)+(2 y+7 r)\)

Short Answer

Expert verified
The sum is \(8y + 2r\).

Step by step solution

01

Identify Like Terms

In the given expression \((6 y-5 r)+(2 y+7 r)\), identify the like terms. These are terms that have the same variable. We have two groups of like terms: the \(y\) terms \((6y\) and \(2y)\) and the \(r\) terms \((-5r\) and \(7r)\).
02

Combine Like Terms

Now, add the coefficients of each group of like terms. For the \(y\) terms: \(6y + 2y = (6 + 2)y = 8y\). For the \(r\) terms: \(-5r + 7r = (-5 + 7)r = 2r\).
03

Write the Simplified Expression

After combining the like terms, we write the simplified expression as a sum of \(y\) and \(r\) terms: \(8y + 2r\). Thus, the sum of the expression is \(8y + 2r\).

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Key Concepts

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

Algebraic Expressions
An algebraic expression is a combination of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. It's a way of representing numbers with symbols and can interpret patterns or relationships in mathematics.
For instance, in the expression \(6y - 5r\),
  • "6y" is a term where 6 is a coefficient and \(y\) is a variable.
  • "-5r" is another term, with -5 as the coefficient and \(r\) being the variable.
The terms are primarily connected through the addition or subtraction of their coefficients. Algebraic expressions are foundational to solving equations and understanding how numbers interact through variables.
Coefficients
Coefficients are numbers that multiply variables in an algebraic expression. They indicate how many times a variable is being added to itself. For example, in the term \(6y\), the coefficient is 6, which means 6 times the variable \(y\).
Understanding coefficients is crucial as it defines:
  • The weight or magnitude of the variable in the expression.
  • How the terms combine when simplifying or solving expressions.
When solving problems, like combining like terms, it鈥檚 essential to adjust and add coefficients correctly to simplify expressions and find meaningful results.
Simplifying Expressions
Simplifying expressions involves combining like terms to form a cleaner, less complex expression. Like terms are terms that have the same variables raised to the same powers, though the coefficients might differ.
For the expression \((6y - 5r) + (2y + 7r)\):
  • You first identify like terms: \(6y\) and \(2y\) as one group, and \(-5r\) and \(7r\) as another.
  • Then, combine their coefficients: \(6+2\) for \(y\) terms gives \(8y\), and \(-5+7\) for \(r\) terms gives \(2r\).
By working through this, the simplified expression becomes \(8y + 2r\). This method makes complex expressions easier to handle and solve in further equations or evaluations.
Prealgebra Concepts
Prealgebra concepts lay the foundation for learning algebra by introducing basic mathematical principles required for higher-level math. It equips students with the ability to understand and manipulate numbers and simple operations within expressions.
Key aspects include:
  • Understanding variables and coefficients and how they are used to represent numbers.
  • Combining like terms to simplify an expression.
  • Basic operations like addition and subtraction are used consistently to rewrite expressions.
Prealgebra provides hands-on practice with these concepts, paving the way for more advanced problems involving equations and inequalities. It鈥檚 like setting the stage for a smooth transition into detailed algebra studies.

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Most popular questions from this chapter

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