Write each expression as a single fraction in simplest form.
Write each expression as a single fraction in simplest form.
- [latex]\displaystyle \dfrac{1}{x} + \dfrac{1}{y}[/latex]
- [latex]\displaystyle 1 - \dfrac{1}{x}[/latex]
- [latex]\displaystyle \dfrac{1}{x - 1} + \dfrac{1}{x + 1}[/latex]
- [latex]\displaystyle x - \dfrac{x}{x + 1}[/latex]
- [latex]\displaystyle \dfrac{\dfrac{y}{x} - \dfrac{x}{y}}{\dfrac{y}{x} + 1}[/latex]
- [latex]\displaystyle \dfrac{1 - \dfrac{x^2}{y^2}}{1 + \dfrac{x^2}{y^2}}[/latex]
- [latex]\displaystyle \dfrac{\dfrac{2a}{x}}{1 - \dfrac{a^2}{x^2}}[/latex]
- [latex]\displaystyle \dfrac{\dfrac{a}{x} + \dfrac{b}{y}}{1 - \dfrac{ab}{xy}}[/latex]
[latex]\underline{\qquad\qquad\qquad\qquad}[/latex]
Algebra Refresher Answers
- [latex]\displaystyle \dfrac{x + y}{xy}[/latex]
- [latex]\displaystyle \dfrac{x - 1}{x}[/latex]
- [latex]\displaystyle \dfrac{2x}{x^2 - 1}[/latex]
- [latex]\displaystyle \dfrac{x^2}{x + 1}[/latex]
- [latex]\displaystyle \dfrac{y - x}{y}[/latex]
- [latex]\displaystyle \dfrac{y^2 - x^2}{y^2 + x^2}[/latex]
- [latex]\displaystyle \dfrac{2ax}{x^2 - a^2}[/latex]
- [latex]\displaystyle \dfrac{ay + bx}{xy - ab}[/latex]
- Recognize identities
- Verify identities
- Rewrite expressions using identities
- Use identities to evaluate expressions
- Solve trigonometric equations
- Given one trig ratio, find the others #59–72
What Is an Identity?
Recall that an equation may be true or false, depending on the values of any variables involved. For example, the equation [latex]x^2 + 3x =10[/latex]
is true only if [latex]x = 2[/latex] or [latex]x = 5{.}[/latex] An equation that is true only for certain values of the variable, and false for others, is called a conditional equation. When you solve a conditional equation, you are finding the values of the variable that make the equation true. Some equations are true for all legitimate values of the variables. Such equations are called identities. Here are some examples of identities.
[latex]3(x + y) =3x + 3y[/latex]
[latex](x + 1)^2 = x^2 + 2x + 1[/latex]
In an identity, the expressions on either side of the equal sign are equivalent expressions, because they have the same value for all values of the variable.
An identity is an equation that is true for all legitimate values of the variables.
Which of the following equations are identities?
- [latex]\displaystyle 3s + 7s = 10s[/latex]
- [latex]\displaystyle 5c(c - 2s) = 5c^2 - 10cs[/latex]
- [latex]\displaystyle 2t - 1 = 3[/latex]
Solution
Many of the algebraic operations you have already learned, such as combining like terms or applying the distributive law, produce equivalent expressions.
- Equation (a) is an identity obtained by combining like terms on the left side.
- Equation (b) is an identity obtained by applying the distributive law on the left side.
- Equation (c) is not an identity, because the equation is true only for [latex]t = 2{.}[/latex]
Checkpoint 5.42.
Which of the following equations are identities?
- [latex]\displaystyle (c - s)(c + s) = c^2 - s^2[/latex]
- [latex]\dis