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Chapter 1: Problem 86

Evaluate each expression when \(x=9\) and \(y=-2 .\) See Example 12 $$ 4 x-10 y $$

Short Answer

Expert verified
The value of the expression is 56.

Step by step solution

01

Substitute the given values.

Start by substituting the given values of \(x\) and \(y\) into the expression. You are given \(x = 9\) and \(y = -2\). Substitute these values into the expression \(4x - 10y\):\[4(9) - 10(-2)\].
02

Simplify the expression.

Calculate each part of the expression. First, compute \(4 \times 9\) which equals \(36\). Then, calculate \(10 \times (-2)\) which equals \(-20\). Substitute these results back into the main expression to get \[36 - (-20)\].
03

Resolve subtraction of a negative

When you subtract a negative number, it is the same as adding its positive. Therefore, \(36 - (-20)\) becomes \(36 + 20\).
04

Final calculation

Add \(36 + 20\) to get the final result. The sum of these two numbers is \(56\).

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Key Concepts

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

Substitution Method
The substitution method is a crucial skill in algebra, especially when you're tasked with evaluating expressions for specific variable values. This method involves replacing unknown variables in an expression with their given numeric values. Let's consider the example where you're given the expression \(4x - 10y\) and asked to evaluate it when \(x = 9\) and \(y = -2\). Here鈥檚 how you use substitution effectively:
  • Identify the variables in the expression. In our example, these are \(x\) and \(y\).
  • Substitute each variable with its given value. Replace \(x\) with \(9\) and \(y\) with \(-2\) to transform the expression into \(4(9) - 10(-2)\).
By filling in the specific values for variables, substitution transforms an algebraic expression into a numerical form that we can then simplify further.
Simplifying Expressions
Simplifying expressions involves combining like terms and calculations to reduce the expression to its simplest form. Once you substitute the variables, simplifying follows naturally. In our expression \(4(9) - 10(-2)\), simplifying begins with clear steps:
  • Perform the multiplications: Start with \(4 \times 9 = 36\). Next, compute \(10 \times (-2) = -20\).
  • Substitute these computations back into the expression, resulting in \(36 - (-20)\).
Simplifying not only clarifies the final result but also ensures accuracy and reinforces your understanding of mathematical operations. This step confirms that you can confidently transform an expression into its most concise numerical form.
Negative Numbers Handling
Working with negative numbers, especially in expressions, can seem daunting at first, but it's all about understanding a few basic rules. In the expression you've worked on, handling negative numbers arises in the subtraction step \(36 - (-20)\). Here鈥檚 how to manage effectively:
  • Understand that subtracting a negative number is equivalent to adding its positive value. So \(36 - (-20)\) becomes \(36 + 20\).
  • Follow this by straightforward addition, which results in \(56\).
Dealing with negatives is an essential algebraic skill, ensuring you're well-prepared to tackle more complex expressions. Mastery in this area allows for smooth progression through all levels of algebra, where sign changes are common.

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Most popular questions from this chapter

Simplify each expression. See Examples \(11,14,\) and \(15 .\) $$ \frac{7}{8} a-\frac{11}{12}-\frac{1}{2} a+\frac{5}{6} $$

Simplify each expression. See Examples \(11,14,\) and \(15 .\) $$ \frac{1}{4}(8 x-4)-\frac{1}{5}(20 x-6 y) $$

Simplify each expression. See Examples I through 11 . $$ \frac{\frac{1}{5} \cdot 20-6}{10+\frac{1}{4} \cdot 12} $$

In your own words, explain why the empty set is a subset of every set.

Evaluate \(12-(5-3)\) and \((12-5)-3 .\) Use these two expressions and discuss whether subtraction is associative.
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