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Algebra 1 HS Mathematics Unit: 02 Lesson: 01 Two-Step Equations KEY The goal of solving an equation is to isolate the variable (get the variable by itself). This is done by performing inverse operations. One-step equations are solved using addition/subtraction properties or multiplication/division properties to isolate the variable. Remember, what you do to one side of an equation, you must do exactly the same to the other side. Two-Step Equations Two-step equations combine the use of addition/subtraction and multiplication/division properties to isolate the variable. • Use the addition/subtraction properties first to move the numeric value or constant. The value on the same side of the equation and farthest away from the variable. • Then, use multiplication/division properties to isolate the variable. Sample Problems I. Solve as indicated. 1. 2x + 7 = 5 Concrete Model Solution Algebraic Solution Graphic Solution, Table Solution Check = + + + + + + + + - = + + + + + + ©2012, TESCCC 05/16/12 page 1 of 11 + y1 = 2x + 7 y2 = 5 Algebra 1 HS Mathematics Unit: 02 Lesson: 01 Two-Step Equations KEY 2. 5x – 7 = –2 Concrete Model Solution Algebraic Solution Graphic Solution, Table Solution Check = + + + + + - = + + + + + + + + + + + + + + + + + + + + + + + ©2012, TESCCC 05/16/12 page 3 of 11 y1 = 5x – 7 y2 = -2 Algebra 1 HS Mathematics Unit: 02 Lesson: 01 Two-Step Equations KEY 3. 1 5 x+3=4 Algebraic Solution 1 5 Graphic Solution, Table y1 = (1/5)x + 3 y2 = 4 x+3=4 Solution Check 1 5 (5) + 3 = 4 1+3=4 4=4 1 5 x+3–3=4–3 1 5 x=1 1 5 ( x) ● 5 = 1 ● 5 x=5 4. 8 – 3x = 14 Algebraic Solution ©2012, TESCCC Graphic Solution, Table 05/16/12 Solution Check page 5 of 11 8 – 3x = 14 8 – 3x – 8 = 14 – 8 -3x = 6 -3 -3 x = -2 y1 = 8 – 3x y2 = 14 8 – 3(-2) = 14 8 + 6 = 14 14 = 14 Algebra 1 HS Mathematics Unit: 02 Lesson: 01 Two-Step Equations KEY II. Formulate an equation for the word problems. Solve the equation algebraically. Write a statement to justify the solution. 5. Twice a number increased by four is fourteen. What is the number? 2x + 4 = 14 x=5 Two times five plus four does equal fourteen. 6. Five less than six times a number is seven. What is the number? 6x – 5 = 7 x=2 Six times two subtract five does equal seven. 7. At the local HUB’s grocery store, the cost of a carton of tamales, c, depends on the number of tamales, t. The cost of the carton is $0.25 and each tamale is $0.50. a. What equation could be used to represent this situation? c = 0.5t + 0.25 b. If a carton contained one dozen tamales, what would be the cost? c = 0.5(12) + 0.25 c = 6.25 c. If the cost of a carton of tamales is $15.25, how many tamales are in the carton? 15.25 = 0.5x + 0.25 x = 30 tamales III. Use the given information to solve for the indicated variable. 8. If (x, -2.8) is a solution to the equation 2x + 4y = 4.5, what is the value of x? 2x + 4(-2.8) = 4.5 2x – 11.2 = 4.5 2x = 15.7 x = 7.85 3 9. If ( 3 3 8 , y) is a solution to the equation 8x = 3y – 7, what is the value of y? 3 8 8( ) = 3y – 7 27 = 3y – 7 34 = 3y 1 11 3 y= ©2012, TESCCC 05/16/12 page 7 of 11 Algebra 1 HS Mathematics Unit: 02 Lesson: 01 Two-Step Equations KEY Practice Problems Solve as indicated. 1. 3x – 3 = 12 Concrete Model Solution Algebraic Solution Graphic Solution, Table Solution Check = + + = + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ©2012, TESCCC 05/16/12 page 9 of 11 y1 = 3x – 3 y2 = 12 2. 1 x +2 =1 3 Algebraic Solution 1 3 1 3 Graphic Solution, Table y1 = (1/3)x + 2 y2 = 1 x+2=1 Solution Check 1 3 (-3) + 2 = 1 -1 + 2 = 1 1=1 x+2–2=1–2 1 3 x = -1 1 3 ( x) ● 3 = -1 ● 3 x = -3 Two-Step Equations KEY Formulate an equation for the word problems. Solve the equation algebraically. Write a statement to justify the solution. Algebra 1 HS Mathematics Unit: 02 Lesson: 01 3. The product of three times a number increased by seven is seventeen. What is the number? 3x + 7 = 17 3x = 10 1 3 x=3 1 3 Three times 3 equals ten. Seven added to ten equals 17. 4. Eleven decreased by twice a number is fifteen. What is the number? 11 – 2x = 15 x = -2 Eleven decreased by 2(-2) or -4, means to add four, so this does equal 15. 5. Surf City charges a base fee of $10 and $25 per hour to rent a surf board. a) What equation could be used to represent the situation? c = 25h + 10 b) If Toni rented a surf board for 5 hours, how much would she have to pay? c = 25(5)+ 10 c = $135 c) Steven paid $185 for renting a surf board from Surf City. For how many hours did Steven rent the surf board? 185 = 25h + 10 h = 7 hours 6. If (x, -4) is a solution to the equation 5x – 4y = 10, what is the value of x? 5x – 4(-4) = 10 5x + 16 = 10 5x = -6 6 1 − or − 1 5 5 x= 7. If (7, y) is a solution to the equation 2x + 3y = 44, what is the value of x? 2(7) + 3y = 44 14 + 3y = 44 3y = 30 y = 10 ©2012, TESCCC 05/16/12 page 11 of 11