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Chapter 2: Problem 11

Simplify each expression as completely as possible. $$(5-3)(x-2)$$

Short Answer

Expert verified
The simplified expression is \2x - 4\.

Step by step solution

01

- Expand the Expression

To simplify the expression \((5-3)(x-2)\), start by expanding it using the distributive property: \[ (5-3)(x-2) = (5-3) \times (x) + (5-3) \times (-2) \]
02

- Simplify Within the Parentheses

Next, simplify inside the parentheses. Calculate \((5-3)\) first: \[ 5-3 = 2 \]
03

- Multiply Out

Now substitute back into the expanded expression: \[ 2 \times x + 2 \times (-2) = 2x - 4 \]
04

- Combine Like Terms

The final step is to combine any like terms. In this case, there are no further like terms to combine. The simplified expression is: \[ 2x - 4 \]

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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 very useful in simplifying algebraic expressions. It allows us to remove parentheses by distributing multiplication over addition or subtraction inside the parentheses. For example, if we have \( a(b + c) \), we can use the distributive property to expand it to \( ab + ac \). This works the same way with subtraction: \( a(b - c) = ab - ac \). In our exercise, we apply this property to look at the expression \( (5-3)(x-2) \). This means we multiply \( (5-3) \) with both \( x \) and \(-2 \).
Combining Like Terms
Once you use the distributive property to expand an expression, the next important step is to combine like terms. Like terms are terms that have the same variables raised to the same power. For instance, in the expression \( 2x - 4 \), \( 2x \) and \(-4 \) are not like terms because one has an \( x \) and the other is a constant. So, they cannot be combined. By combining like terms, you simplify the expression further. Make sure you group terms correctly and perform the addition or subtraction as needed.
Basic Algebra
Basic algebra involves understanding and applying fundamental principles to manipulate algebraic expressions. This includes operations like addition, subtraction, multiplication, and division of variables and constants. A key skill is the ability to simplify expressions by performing operations correctly, such as removing parentheses through the distributive property and combining like terms. In the given exercise, we began with an expression involving parentheses and variables, distributed the terms, and simplified step-by-step until we reached \( 2x - 4 \). Such practice with basic algebra builds a foundation for solving more complex algebraic equations and problems.

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Most popular questions from this chapter

Evaluate the given expression for \(x=2, y=-3,\) and \(z=-4.\) $$(-x)^{2}-z$$

Simplify each expression as completely as possible. $$-a^{2}(3 a-7)-2 a\left[a^{2}-4(a-2)\right]$$

Evaluate the given expression for \(x=2, y=-3,\) and \(z=-4.\) $$\frac{x y z}{z-x}$$

Explain what is wrong with each of the following: (a) \(3 \cdot 2^{2} \stackrel{?}{=} 6^{2} \stackrel{?}{=} 36\) (b) \(3 x^{4} \stackrel{?}{=} 3 x \cdot 3 x \cdot 3 x \cdot 3 x\) (c) \(5^{4} \cdot 2 \stackrel{?}{=} 20 \cdot 2 \stackrel{?}{=}40\) (d) \(\left(3 x^{3}\right)^{2} \stackrel{?}{=} 6 x^{6}\) (e) \(-3^{4} \stackrel{?}{=}(-3) \cdot(-3) \cdot(-3)(-3) \stackrel{?}{=} 81\) (f) \(x^{4}+x^{5} \stackrel{?}{=} x^{9}\) (g) \(x^{2} \cdot x^{7} \stackrel{?}{=} x^{14}\) (h) \(\left(x^{2}\right)^{4} \stackrel{?}{=} x^{6}\)

Evaluate the given expression for \(x=2, y=-3,\) and \(z=-4.\) $$2 z^{2}-5 z-4$$
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