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Solving Equations With Variables on Both Sides & Special Cases Algebra 1 Glencoe McGraw-Hill JoAnn Evans Some equations have variables on both sides of the equal sign. To solve such equations, collect like variables on the same side. In the equation shown below, the term 9x has a greater coefficient than the term 4x. You can avoid a negative coefficient by collecting variables on the right. 4x 12 9x 8 4x 4x 12 5x 8 8 8 20 5x 5 5 4x 4x 12 9x 8 9x 9x 5x 12 8 12 12 But it 5x 20 works 5 5 this way x4 too. Solve the equation by collecting like variables on one side. Check the solution. 7x 19 2x 55 2x 2x 9x 19 55 19 19 9x 36 9 9 x4 7 4 19 2 4 55 28 19 8 55 Solve the equation by collecting like variables on one side. Check the solution. 4 1 x 3x 2 x 1 4 4x 3x 2x 2 4 x 2x 2 2x 2x 4 x 2 4 4 x 6 4 1 6 3 6 2 6 1 4 7 18 2 5 28 18 10 Solve the equation by collecting like variables on one side. Check the solution. 1 12x 16 10 3 x 2 4 1 12x 16 10 3 x 2 4 3x 4 10 3x 6 3x 4 16 3x 3x 3x 6x 4 16 4 4 6x 12 x 2 Check the solution with mental math. Try this one on your own: 7 2 x 5x 14 7x 5x 7x 7x 7 7 7 2 5 6 6 12 7 7 7 5 6 6 6 5 7 7 5 6 6 35 35 6 6 14 12x 12 12 7 x 6 Don’t be reluctant to check a fractional solution. It’s just arithmetic! A special type of equation 3 4 x 2x 12 x 12 3x 3x 12 3x 12 3x 12 3x 3x 12 12 commutative property of addition When you try to solve an identity, you end up with a statement that is always true. When the two sides of an equation are identical it’s called an identity. What happens if you continue solving? An identity will have INFINITELY MANY SOLUTIONS. When the two sides of an equation are identical, any number substituted in as a solution will make a true statement. All real numbers are solutions of an identity. Write: All Real Numbers Another special type of equation 8x 4 2 8x 8x 4 2 8x 8x 6 8x 8x 8x Untrue statement! 6 0 When all variable terms cancel out and what’s left is an untrue statement of equality, there is no solution to the equation. Write: No Solution Solve the equation, if possible. Determine whether it has one solution, no solution, or is an identity and thus all real numbers are solutions. 5(x 3) 3x 15 2x 5x 15 5x 15 5x 5x 15 15 It is an identity. All real numbers are solutions of this equation. 5(x 3) 3x 10 2x 5x 15 5x 10 5x 5x 15 10 An untrue statement is left, so there is no solution to this equation. No value of the variable will make the equation true. 5(x 3) 3x 5 x 5x 15 4x 5 4x 4x x 15 5 15 15 x 10 This equation has one solution. x = -10 1 6(2m 7) (18 12m) 3 12m 42 6 4m 4m 4m 8m 42 6 42 42 8m 48 8 8 m6 One solution! 8(5c 2) 10(32 4c) 40c 16 320 40c 40c 40c 1 4( y 20) (20y 400) 5 4 y 80 4 y 80 16 320 No solution! All real numbers! Extra practice problems: 1. 2. 3. 4. 2 2 3x 3 3 x 4 5x 9 3x 13 2 11 x 14 6 3n 4n n 3y 2 y 19 9y 3 9 y 1. x 1 2. all real numbers 3. no solution 4. y 1