Equations To solve an algebraic equation with one variable, find the value of the unknown variable. Rules for Working with Equations 1. The equal sign separates an equation into two sides. 2. Whenever an operation is performed on one side, the same operation must be performed on the other side. 3. To solve an equation, first move all of the variables to one side and all of the numbers to the other. Then simplify until only one variable (with a coefficient of 1) remains on one side and one number remains on the other side. CHAPTER Algebra Review This chapter reviews key skills and concepts of algebra that you need to know for the SAT. Throughout the chapter are sample questions in the style of SAT questions. Each sample SAT question is followed by an explanation of the correct answer. 6 69 Example 7x Ϫ 11 ϭ 29 Ϫ 3x Move the variables to one side. 7x Ϫ 11 ϩ 3x ϭ 29 Ϫ 3x ϩ 3x Perform the same operation on both sides. 10x Ϫ 11 ϭ 29 Now move the numbers to the other side. 10x Ϫ 11 ϩ 11 ϭ 29 ϩ 11 Perform the same operation on both sides. 10x ϭ 40 Divide both sides by the coefficient. ᎏ 1 1 0 0 x ᎏ ϭ ᎏ 4 1 0 0 ᎏ Simplify. x ϭ 4 Practice Question If 13x Ϫ 28 ϭ 22 Ϫ 12x, what is the value of x? a. Ϫ6 b. Ϫ ᎏ 2 6 5 ᎏ c. 2 d. 6 e. 50 Answer c. To solve for x: 13x Ϫ 28 ϭ 22 Ϫ 12x 13x Ϫ 28 ϩ 12x ϭ 22 Ϫ 12x ϩ 12x 25x Ϫ 28 ϭ 22 25x Ϫ 28 ϩ 28 ϭ 22 ϩ 28 25x ϭ 50 x ϭ 2 Cross Products You can solve an equation that sets one fraction equal to another by finding cross products of the fractions. Find- ing cross products allows you to remove the denominators from each side of the equation by multiplying each side by a fraction equal to 1 that has the denominator from the opposite side. Example ᎏ a b ᎏ ϭ ᎏ d c ᎏ First multiply one side by ᎏ d d ᎏ and the other by ᎏ b b ᎏ . The fractions ᎏ d d ᎏ and ᎏ b b ᎏ both equal 1, so they don’t change the equation. ᎏ a b ᎏ ϫ ᎏ d d ᎏ ϭ ᎏ d c ᎏ ϫ ᎏ b b ᎏ ᎏ a bd d ᎏ ϭ ᎏ b b d c ᎏ The denominators are now the same. Now multiply both sides by the denominator and simplify. bd ϫ ᎏ a bd d ᎏ ϭ bd ϫ ᎏ b b d c ᎏ ad ϭ bc The example above demonstrates how finding cross products works. In the future, you can skip all the middle steps and just assume that ᎏ a b ᎏ ϭ ᎏ d c ᎏ is the same as ad ϭ bc. –ALGEBRA REVIEW– 70 Example ᎏ 6 x ᎏ ϭ ᎏ 1 3 2 6 ᎏ Find cross products. 36x ϭ 6 ϫ 12 36x ϭ 72 x ϭ 2 Example ᎏ 4 x ᎏ ϭ ᎏ x ϩ 16 12 ᎏ Find cross products. 16x ϭ 4(x ϩ 12) 16x ϭ 4x ϩ 48 12x ϭ 48 x ϭ 4 Practice Question If ᎏ 9 y ᎏ ϭ ᎏ y 1 Ϫ 2 7 ᎏ , what is the value of y? a. Ϫ28 b. Ϫ21 c. Ϫ ᎏ 6 1 3 1 ᎏ d. Ϫ ᎏ 7 3 ᎏ e. 28 Answer b. To solve for y: ᎏ 9 y ᎏ ϭ ᎏ y 1 Ϫ 2 7 ᎏ Find cross products. 12y ϭ 9(y Ϫ 7) 12y ϭ 9y Ϫ 63 12y Ϫ 9y ϭ 9y Ϫ 63 Ϫ 9y 3y ϭϪ63 y ϭϪ21 Checking Equations After you solve an equation, you can check your answer by substituting your value for the variable into the orig- inal equation. Example We found that the solution for 7x Ϫ 11 ϭ 29 Ϫ 3x is x ϭ 4. To check that the solution is correct, substitute 4 for x in the equation: 7x Ϫ 11 ϭ 29 Ϫ 3x 7(4) Ϫ 11 ϭ 29 Ϫ 3(4) 28 Ϫ 11 ϭ 29 Ϫ 12 17 ϭ 17 This equation checks, so x ϭ 4 is the correct solution! –ALGEBRA REVIEW– 71 Equations with More Than One Variable Some equations have more than one variable. To find the solution of these equations, solve for one variable in terms of the other(s). Follow the same method as when solving single-variable equations, but isolate only one variable. Example 3x ϩ 6y ϭ 24 To isolate the x variable, move 6y to the other side. 3x ϩ 6y Ϫ 6y ϭ 24 Ϫ 6y 3x ϭ 24 Ϫ 6y ᎏ 3 3 x ᎏ ϭ ᎏ 24 Ϫ 3 6y ᎏ Then divide both sides by 3, the coefficient of x. x ϭ 8 Ϫ 2y Then simplify. The solution is for x in terms of y. Practice Question If 8a ϩ 16b ϭ 32, what does a equal in terms of b? a. 4 Ϫ 2b b. 2 Ϫ ᎏ 1 2 ᎏ b c. 32 Ϫ 16b d. 4 Ϫ 16b e. 24 Ϫ 16b Answer a. To solve for a in terms of b: 8a ϩ 16b ϭ 32 8a ϩ 16b Ϫ 16b ϭ 32 Ϫ 16b 8a ϭ 32 Ϫ 16b ᎏ 8 8 a ᎏ ϭ ᎏ 32 Ϫ 8 16b ᎏ a ϭ 4 Ϫ 2b 72 Special Tips for Checking Equations on the SAT 1. If time permits, check all equations. 2. For questions that ask you to find the solution to an equation, you can simply substitute each answer choice into the equation and determine which value makes the equation correct. Begin with choice c. If choice c is not correct, pick an answer choice that is either larger or smaller. 3. Be careful to answer the question that is being asked. Sometimes, questions require that you solve for a variable and then perform an operation. For example, a question may ask the value of x Ϫ 2. You might find that x = 2 and look for an answer choice of 2. But the question asks for the value of x Ϫ 2 and the answer is not 2, but 2 Ϫ 2. Thus, the answer is 0. Monomials A monomial is an expression that is a number, a variable, or a product of a number and one or more variables. 6 y Ϫ5xy 2 19a 6 b 4 Polynomials A polynomial is a monomial or the sum or difference of two or more monomials. 7y 5 Ϫ6ab 4 8x ϩ y 3 8x ϩ 9y Ϫ z Operations with Polynomials To add polynomials, simply combine like terms. Example (5y 3 Ϫ 2y ϩ 1) ϩ (y 3 ϩ 7y Ϫ 4) First remove the parentheses: 5y 3 Ϫ 2y ϩ 1 ϩ y 3 ϩ 7y Ϫ 4 Then arrange the terms so that like terms are grouped together: 5y 3 ϩ y 3 Ϫ 2y ϩ 7y ϩ 1 Ϫ 4 Now combine like terms: Answer: 6y 3 ϩ 5y Ϫ 3 Example (2x Ϫ 5y ϩ 8z) Ϫ (16x ϩ 4y Ϫ 10z) First remove the parentheses. Be sure to distribute the subtraction correctly to all terms in the second set of parentheses: 2x Ϫ 5y ϩ 8z Ϫ 16x Ϫ 4y ϩ 10z Then arrange the terms so that like terms are grouped together: 2x Ϫ 16x Ϫ 5y Ϫ 4y ϩ 8z ϩ 10z –ALGEBRA REVIEW– 73 Three Kinds of Polynomials ■ A monomial is a polynomial with one term, such as 5b 6 . ■ A binomial is a polynomial with two unlike terms, such as 2x + 4y. ■ A trinomial is a polynomial with three unlike terms, such as y 3 + 8z Ϫ 2. Now combine like terms: Ϫ14x Ϫ 9y ϩ 18z To multiply monomials, multiply their coefficients and multiply like variables by adding their exponents. Example (Ϫ4a 3 b)(6a 2 b 3 ) ϭ (Ϫ4)(6)(a 3 )(a 2 )(b)(b 3 ) ϭϪ24a 5 b 4 To divide monomials, divide their coefficients and divide like variables by subtracting their exponents. Example ᎏ 1 1 0 5 x x 5 4 y y 7 2 ᎏ ϭ ( ᎏ 1 1 0 5 ᎏ )( ᎏ x x 5 4 ᎏ )( ᎏ y y 7 2 ᎏ ) ϭ ᎏ 2x 3 y 5 ᎏ To multiply a polynomial by a monomial, multiply each term of the polynomial by the monomial and add the products. Example 8x(12x Ϫ 3y ϩ 9) Distribute. (8x)(12x) Ϫ (8x) (3y) ϩ (8x)(9) Simplify. 96x 2 Ϫ 24xy ϩ 72x To divide a polynomial by a monomial, divide each term of the polynomial by the monomial and add the quotients. Example ᎏ 6x Ϫ 1 6 8y ϩ 42 ᎏ ϭ ᎏ 6 6 x ᎏ Ϫ ᎏ 1 6 8y ᎏ ϩ ᎏ 4 6 2 ᎏ ϭ x Ϫ 3y ϩ 7 Practice Question Which of the following is the solution to ᎏ 1 2 8 4 x x 8 3 y y 5 4 ᎏ ? a. ᎏ 4x 3 5 y ᎏ b. ᎏ 18 2 x 1 4 1 y 9 ᎏ c. 42x 11 y 9 d. ᎏ 3x 4 5 y ᎏ e. ᎏ x 6 5 y ᎏ Answer d. To find the quotient: ᎏ 1 2 8 4 x x 8 3 y y 5 4 ᎏ Divide the coefficients and subtract the exponents. ᎏ 3x 8 Ϫ 4 3 y 5 Ϫ 4 ᎏ ᎏ 3x 4 5 y 1 ᎏ ϭ ᎏ 3x 4 5 y ᎏ –ALGEBRA REVIEW– 74 . ϭ bc. –ALGEBRA REVIEW 70 Example ᎏ 6 x ᎏ ϭ ᎏ 1 3 2 6 ᎏ Find cross products. 36x ϭ 6 ϫ 12 36x ϭ 72 x ϭ 2 Example ᎏ 4 x ᎏ ϭ ᎏ x ϩ 16 12 ᎏ Find cross products. 16x ϭ 4(x ϩ 12) 16x ϭ 4x ϩ 48 12x. 8a ϩ 16b ϭ 32, what does a equal in terms of b? a. 4 Ϫ 2b b. 2 Ϫ ᎏ 1 2 ᎏ b c. 32 Ϫ 16b d. 4 Ϫ 16b e. 24 Ϫ 16b Answer a. To solve for a in terms of b: 8a ϩ 16b ϭ 32 8a ϩ 16b Ϫ 16b ϭ 32 Ϫ 16b 8a. isolate only one variable. Example 3x ϩ 6y ϭ 24 To isolate the x variable, move 6y to the other side. 3x ϩ 6y Ϫ 6y ϭ 24 Ϫ 6y 3x ϭ 24 Ϫ 6y ᎏ 3 3 x ᎏ ϭ ᎏ 24 Ϫ 3 6y ᎏ Then divide both sides by 3, the