Problem 83 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(77-96,\) simplify each algebraic expression. $$-5 x+x$$

Step-by-Step Solution

Answer

The simplified form of the expression \(-5x + x\) is \(-4x\).

Step 1: Combine like terms

In the given expression, we have two terms, \(-5x\) and \(x\), that are like terms, meaning they have the same variable raised to the same power. When you combine these terms, you add or subtract the coefficients (the numbers in front of the variable). For the expression \(-5x + x\), we add the coefficients. The term \(x\) does not have an explicit coefficient but, when not given, the coefficient is understood to be 1. So, the expression simplifies to \(-5x + x = -5x + 1x\).

Step 2: Simplify

Now add together the coefficients -5 and 1. This simplification gives \(-5x + 1x = -4x\).

Key Concepts

Combining Like Terms

Imagine you have a collection of apples and you're trying to count them. Some are green and some are red. Only by combining the apples that are alike can you accurately know how many of each you have. Similarly, in algebra, **combining like terms** allows us to simplify expressions by grouping terms that have the same variable part.To do this:

• Look for terms that have identical variable parts. In \[-5x + x\], both terms have 'x', so they are like terms.
• Add or subtract their coefficients. Remember, coefficients are the numbers in front of the variables.
• If a term doesn't clearly show a coefficient, assume it's 1. For \(x\), the hidden coefficient is 1.

In our example, combining like terms gives us \[-5x + 1x = -4x\]. This results in a cleaner expression that's much easier to work with.

Coefficients

Coefficients in algebra are the numbers that sit right next to the variables. They do the heavy lifting for us in calculations, as they tell us how many of each variable are being considered.In the expression \[-5x + x\], the coefficients are -5 and 1 (though 1 is invisible). Here's what you should know about coefficients:

• The sign (+ or -) before a number is part of the coefficient and it affects how terms are combined. If you have -5 and 1 as coefficients, you subtract them as \(-5 + 1 = -4\).
• A zero coefficient means the term effectively disappears from the expression (e.g., \(0x = 0\)).

By understanding coefficients, you gain control over simplifying expressions. They act as the numerical force to unite our like terms.

Variables

At the core of all algebraic expressions, **variables** serve as placeholders for unknown or changing numbers. They allow practical mathematics to model real-world situations and solve problems. In our problem, 'x' functions as the variable. Here are some key insights:

• Variables can represent a single number or a range of numbers. They're versatile and can adapt to different contexts depending on their equation.
• When you combine like terms, you're grouping the same variables together, which makes solving equations more straightforward.
• Variables are usually letters, but they can be any symbol that hasn't been defined as something else.

Even when you might not know exactly what the variable is equal to, understanding how it interacts with coefficients and other elements in an equation helps simplify algebraic expressions.