Problem 71 - Algebra 1

Question

As parts (a) and (b) of Example 3 show, it is sometimes easier to evaluate an expression by simplifying it before substituting, and sometimes easier if you substitute for the variable first. a. Write an expression that is easier to evaluate if you simplify before substituting 12 for \(x\) b. Write an expression that is easier to evaluate if you substitute 12 for \(x\) first.

Step-by-Step Solution

Answer

a. An expression that is easier to evaluate by simplifying before substituting is \((x^{2} - 1) / (x - 1)\) \nb. An expression that is easier to evaluate by substituting first is \((x + 3)^2\)

Step 1: Generate an expression that is easier to evaluate by simplification first

Consider the expression \((x^{2} - 1) / (x - 1)\). If \(x = 12\) is substituted immediately, the division operation makes it complex. However, by simplifying first using difference of squares to rewrite the expression as \((x - 1)(x + 1) / (x - 1)\), the term \((x - 1)\) in the numerator and the denominator cancel each other, simplifying the expression to \(x + 1\). Following this, we can easily substitute \(x = 12\), which gives us 13 as the result.

Step 2: Generate an expression that is easier to evaluate by substitution first

Consider the expression \((x + 3)^2\). If \(x = 12\) is substituted immediately, we get \((12 + 3)^2 = 15^2 = 225\). This is simpler than first expanding the expression to \(x^2 + 6x + 9\) and then substituting \(x = 12\), which will require more arithmetic operationsvto arrive at the same result.

Key Concepts

Substitution Method

The substitution method is a valuable tool in mathematics, particularly in simplifying and evaluating algebraic expressions. It essentially involves replacing a variable with a specific value. This method can be particularly useful when dealing with expressions where substituting the value directly makes the calculation simpler and less complex, avoiding unnecessary algebraic manipulations.

Consider the expression

• \((x + 3)^2\)

If you directly substitute, letting \(x = 12\), we calculate

• \((12 + 3)^2 = 15^2 = 225\)

The arithmetic is straightforward and less error-prone. By substituting first, it minimizes steps and reduces the chance of arithmetic errors. The key takeaway from using substitution is recognizing when a direct replacement can significantly streamline the solving process, particularly when it reduces complexity or the number of calculations needed.

Difference of Squares

The difference of squares is a powerful algebraic technique that allows you to factor and simplify expressions where you have two perfect squares being subtracted. It applies the identity

• \(a^2 - b^2 = (a - b)(a + b)\)

This identity not only helps in simplifying expressions but also in performing operations like division or multiplication when these appear troublesome.

For instance, take the expression

• \(x^2 - 1\)

This can be factored using the difference of squares as

• \((x - 1)(x + 1)\)

When divided by

• \(x - 1\)

, it allows you to cancel

• \((x - 1)\)

thereby simplifying the expression to just

• \(x + 1\)

This simplification makes subsequent calculations much easier. Using the difference of squares creates opportunities to break down complex problems into simpler components by factoring, which can save quite a bit of time and makes evaluation clearer.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a fundamental skill in algebra that involves reducing expressions into their simplest forms without changing their value. This process can involve various algebraic techniques such as factoring, combining like terms, and using basic identities or rules like the difference of squares. The main aim is to make expressions easier to manage or evaluate.

Let's consider the expression

• \((x^2 - 1) / (x - 1)\)

If simplified before substitution, by using the difference of squares, it becomes

• \(x + 1\)

Since we canceled the larger component, dealing with it becomes much less cumbersome. This allows for the easy substitution of any value of \(x\) into the expression. In our example, substituting \(x = 12\) directly into the simplified expression results in much easier arithmetic, giving us 13.

Understanding and mastering different methods to simplify expressions help effectively solve algebraic problems. It also allows for better insight into the structure of mathematical expressions, leading to smarter and optimized problem-solving strategies.