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Chapter 5: Problem 102

Subtract. $$ (9 r-5 s-t)-(7 r-5 s+3 t) $$

Short Answer

Expert verified
2r - 4t

Step by step solution

01

Distribute the Negative Sign

Rewrite the subtraction expression by distributing the negative sign to each term inside the second set of parentheses: (9r - 5s - t) - (7r - 5s + 3t) becomes (9r - 5s - t) + (-7r + 5s - 3t)
02

Combine Like Terms

Group and combine the like terms separately for each variable. - For the terms with r: 9r - 7r = 2r - For the terms with s: -5s + 5s = 0s (these cancel out) - For the terms with t: -t - 3t = -4t
03

Write the Final Expression

After combining the like terms, write down the simplified expression: 2r - 4t.

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Key Concepts

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

Distributive Property
The distributive property is an important concept in algebra that helps simplify expressions and solve equations. It states that for any numbers or variables a, b, and c: \[ a(b + c) = ab + ac \].
In the given exercise, we need to distribute the negative sign across the terms inside the second set of parentheses. This step is crucial because it changes the signs of all the terms within:
(9r - 5s - t) - (7r - 5s + 3t) turns into (9r - 5s - t) + (-7r + 5s - 3t).
This effectively means we're subtracting each term in the second set of parentheses from the terms in the first set. Distributing correctly helps avoid mistakes in sign, which is essential for getting the right solution.
Combining Like Terms
Combining like terms is another vital technique in algebra. It allows us to simplify expressions by adding or subtracting terms with the same variable component. After distributing the negative sign, the expression becomes: (9r - 5s - t) + (-7r + 5s - 3t).
Next, we group and combine the like terms:
  • Terms with r: 9r and -7r
  • Terms with s: -5s and +5s
  • Terms with t: -t and -3t
When we combine these, we get: 9r - 7r = 2r, -5s + 5s = 0 (canceling out), and -t - 3t = -4t. So, the simplified expression is: 2r - 4t. Grouping and combining like terms reduces the complexity of the expression, making it easier to handle in future steps.
Linear Equations
Linear equations are equations of the first degree, meaning they involve variables raised only to the power of one. The simplified expression we obtained, 2r - 4t, represents a linear equation. Linear equations are fundamental in algebra because they often model real-world situations where the relationship between variables is proportional. To solve more complex problems involving linear equations:
  • Identify and isolate the variable terms
  • Combine like terms
  • Maintain equality by performing the same operation on both sides of the equation
. By mastering the handling of linear equations, students can solve various mathematical problems efficiently. Remember, keep the variables and coefficients clearly defined, and always check your work for accuracy.

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Most popular questions from this chapter

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