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Chapter 4: Problem 10

Let \(a=3, b=-2, c=-1,\) and \(d=2 .\) Evaluate each expression. $$a c-b d$$

Short Answer

Expert verified
The evaluated expression \(ac - bd\) is 1.

Step by step solution

01

Substitute the values

The values for \(a, b, c,\) and \(d\) are provided. Substitute these values into the expression. Thus, \(a c-b d\) becomes \(3*(-1) - (-2)*2\).
02

Perform the multiplication operations

By following the order operations, perform the multiplication first. The expression \(3*(-1) - (-2)*2\) becomes \(-3 - (-4)\).
03

Solve the subtraction operation

The expression now is \(-3 - (-4)\), which simplifies to \(-3 + 4\). Completing this operation gives you the final solution.

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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 mathematical phrase that includes numbers, variables, and arithmetic operations. The expression can be as simple as a single variable or a more complex one involving multiple operations. In the expression given, \(a c - b d\), \(a\), \(b\), \(c\), and \(d\) are variables, and the expression involves multiplication and subtraction operations among these variables.

Important points about algebraic expressions include:
  • Variables can represent different numerical values, and their substitution alters the expression's outcome.
  • Algebraic expressions do not have equality signs (like equations) unless they form an equation.
  • They are used to convey relationships between different quantities.
Understanding how to manipulate these expressions is fundamental in algebra to solve problems or simplify expressions.
Order of Operations
To evaluate or simplify algebraic expressions properly, one must follow the order of operations, a set of rules determining the sequence to perform different mathematical operations. The commonly used acronym is PEMDAS:
  • P for Parentheses - Perform operations inside parentheses first.
  • E for Exponents - Resolve any exponents next.
  • M/D for Multiplication/Division - Solve these from left to right as they appear.
  • A/S for Addition/Subtraction - Lastly, perform these operations from left to right.
In our example, the expression \(3*(-1) - (-2)*2\), we followed the order of operations:

First, the multiplication \(3*(-1)\) yields \(-3\) and \((-2)*2\) gives \(-4\). Then, continuing with the subtraction step, we handle the expression \(-3 - (-4)\) by transforming it to \(-3 + 4\), leading to the final result.
Variables Substitution
Substituting variables is a vital step when evaluating algebraic expressions. When a value is given for a variable, it replaces the variable throughout the expression, allowing the expression to be simplified to a numerical value. In this example, we substituted \(a = 3, b = -2, c = -1,\) and \(d = 2\) into the expression \(a c - b d\).

To substitute variables effectively:
  • Identify each variable in the expression and the value it should take.
  • Replace every instance of the variable in the expression with its corresponding value.
  • Simplify the modified expression using the order of operations until you reach a numeric answer.
This approach simplifies complex expressions into workable arithmetic, making problem-solving more straightforward and manageable.

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Most popular questions from this chapter

Perform the operations. $$3\left(x y^{2}+y^{2}\right)-2\left(x y^{2}-4 y^{2}+y^{3}\right)+2\left(y^{3}+y^{2}\right)$$

Perform the operations. $$-8(x-y)+11(x-y)$$

Perform each division. Assume no division by \(0 .\) $$\frac{x^{3}+27}{x+3}$$

Let \(P(x)=3 x-5 .\) Find each value. $$\text { If } P(x)=x^{23}+5 x^{2}+73 \text { and } Q(x)=x^{23}+4 x^{2}+73, \text { find } P(7)-Q(7)$$

Simplify or solve as appropriate. $$(2 s-3)(s+2)=(2 s+1)(s-3)$$
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