Solving Equations An equation is an algebraic expression that says two values are equal. To solve an equation, all the numbers need to be on one side of the equation and the variables need to be on the other side. To accomplish this, remember, whatever you do to one side of the equation, you must do to the other side. The process you will use to separate the numbers from the variables is called inverse operations. An inverse operation is the opposite of the original operation in the equation. Instead of adding, you will subtract; instead of multiplying, you will divide. Look at the equation below: x + 23 = 52 http://bit.ly/hippo_alg6 To solve this equation, you must first look at the addition sign and the number to be added. Since the equation tells you to add, you would do the opposite, or subtract, 23 from both sides of the equation. x + 23 − 23 = x, and 52 − 23 = 29, so the solution is x = 29. http://bit.ly/hippo-alg7 Solving Equations with Variables on Both Sides of the Equal Sign An equation that has variables on both sides of the equal sign is solved in much the same manner as one in which there is a variable on only one side. It will, however, take three steps to solve this type of equation. Look at the example below: 6x + 2 = 5x + 5 http://bit.ly/hippo_alg2 First, use the inverse to get all the unknowns on one side of the equation. 6x + 2−5x = 5x− 5x + 5 x + 2 = 5 Now, use the inverse again to move the 2 to the other side of the equation.
x + 2 − 2 = 5 − 2 x = 3
Practice Solve the following equations 1. 18 = 6 + x 2. y – 22 = 68 3. 3c = 36 4. = 84 5. 3x + 2 = 20 6. 3a + 4 = 40 7. = 12 8. 3y + 7 = 37 9. 2(x + 5)= 3(x − 1) 10. x = 12 Answers 1. 12 2. 90 3. 12 4. 588 5. 6
6. 12 7. 24 8. 10 9. 13 10. 18
Test Yourself 1. Simplify 6x − x + 8x (1) −13x (2) 2x (3) 3x (4) 5x (5) 13x 2. Find the value of −6a + a 2 + 3(a + 6) if a = 3 (1) −18 (2) 9 (3) 18 (4) 27 (5) 28 3. Solve for x: 3(x + 6) = 2(x − 1) (1) −20 (2) −16 (3) 7 (4) 8 (5) 16 4. Solve for w: 3w + 5w = 40 (1) 5 (2) 6
(3) 8 (4) 9 (5) 10 5. After picking berries, Susan had 6 more pints of berries than Jan. Together they had 28 pints. How many pints did Jan have? (1) 6 (2) 9 (3) 11 (4) 21 (5) 28 6. x + y + 3z + 2x − z = (1) x + 3y + z (2) 3x + y + 2z (3) 2x + 3z + y (4) 2x + 3y + z (5) 2x + y + z 7. Manuel earns x dollars in one day. His brother earns y dollars in one day. Which expression shows how much they earn together in 15 days? (1) x + y (2) 15x + y (3) 15(x + 2y) (4) 15x + 15y (5) 15x − y 8. Find the value of x in the equation x + 3x + (x + 3) = 18
(1) 2 (2) 3 (3) 4 (4) 5 (5) 6 9. After a purchase of 35 dollars, Joseph had 75 dollars left. Which equation shows how much money Joseph had originally? (1) 35 + d = 75 (2) 35 − d = 75 (3) 75 + 35 = d (4) 75 − d = 35 (5) 75 − 35 = d 10. Sarah has 25 dollars. She buys 2 books that cost $7.50 each. Which expression represents the amount of money Sarah has left after her purchase? (1) x = 25 − 2(7.50) (2) x = 25 − 7.50 (3) x = 2(7.50) − 25 (4) 2x = 25 − 7.50 (5) Not enough information is provided. 11. Simplify 3x 2y + 2x 2y (1) 5x 2y (2) 5x 2y 2 (3) 5x 4y 2
(4) 6x 2y (5) 6x 4y 12. (−3x 4 )(−4x 2 ) = (1) −12x 6 (2) −12x 2 (3) 12x 2 (4) 12x 6 (5) 12x 8 13. = (1) −2a 2b (2) −2ab (3) 2ab (4) 2a 2b (5) 3ab 14. Find the value of 5x − x 2 + 5(x − 4) if x = 6 (1) 4 (2) 6 (3) 20 (4) 28 (5) 44 15. Solve for x: 2x − 7 = 3(x + 4) (1) −19
(2) −11 (3) 5 (4) 11 (5) 19 16. Solve for z: − 5 = 4 (1) 4 (2) 4.5 (3) 7 (4) 9 (5) 18 17. Solve for q: 4q − 5q = 10 (1) −10 (2) −5 (3) 5 (4) 9 (5) 10 18. Shawn has 22 fewer baseball cards than Janet; together they have 108 cards. How many cards does Janet have? (1) 26 (2) 34 (3) 54 (4) 65 (5) 78
19. 2x − y + 4z − 3x + 3y = (1) −x + 2y + 4z (2) −x − 2y + 4z (3) x + 2y + 4z (4) x − 2y + 4z (5) 2x + 2y + 4z 20. Danielle jogs x miles per week; her friend Alex jogs 2 miles less per week than Danielle. Which expression shows how many miles both of them together jog in 12 weeks? (1) x + 2 (2) x − 2 (3) 12(x − 2) (4) 12x − 2 (5) 12(2x − 2) 21. Find the value of x 2 if x = 3. (1) 4 (2) 6 (3) 9 (4) 16 (5) 18 22. Solve for x: 3x + 12 = 4 − 5x (1) −17 (2) −7 (3) −3
(4) −1 (5) 7 23. 28 students in gym class are split into 4 teams to play volleyball. Which expression shows the number of students on each team? (1) T = 28 − 4 (2) T = 28(4) (3) T = (4) 28 = (5) 28T = 4 24. Justin has d dollars in his wallet and then gets $60 more from an ATM. Now he has $103. Which expression shows how much money he had originally? (1) 60d = 103 (2) d − 103 = 60 (3) 60 + 103 = d (4) d + 60 = 103 (5) 60 = d + 103 25. There are 5 more girls than boys in Mr. Green’s class, and there are 37 students in the class as a whole. How many boys are in the class? HINT: Let x = the number of boys in the class and x + 5 = the number of girls. (1) 5 (2) 10 (3) 16 (4) 17 (5) 22
Answers 1. (5) 6x − x + 8x = 5x + 8x = 13x 2. (3) 3. (1) 4. (1) 5. (3) Let x represent the number of pints Jan has. 6. (2) 7. (4) 15x (Manuel’s earnings) + 15y (his brother’s earnings) = the total they will earn together in 15 days. The correct expression is 15x + 15y.
8. (2) 9. (3) 75 + 35 = d The amount of money left plus the amount he spent would equal the total. 10. (1) x = 25 − 2(7.50). The original amount is 25 dollars. Subtract the amount Sarah spent on the books (2)(7.50), to get the remainder. 11. (1) 3x 2y + 2x 2y = 5x 2y. When adding or subtracting in expressions, the exponents remain the same. 12. (4) (−3x 4 )(−4x 2 ) = 12x 6 . When multiplying in expressions, the exponents are added together. 13. (3) = 2ab. When multiplying in expressions, the exponents are subtracted. 14. (1) Plug in 6 for x: 15. (1) 16. (5)
17. (1) 18. (4) Let x represent the number of baseball cards Shawn has. 19. (1) 20. (5) In one week, the two of them together jog x + (x − 2) miles, or 2x − 2 miles; in 12 weeks they jog 12 times that far, or 12(2x − 2) miles. 21. (2) Plug in 3 for x: 22. (4) 23. (3) To split the class into 4 teams, divide 28 by 4:
24. (4) d + 60 = 103. The amount Justin started out with plus the amount he took out from the ATM equals the new amount. 25. (3)