Solving Equations with Variables on Both Sides Essential

Solving Equations with Variables on Both Sides Essential Question? How can you solve equations with variables on both sides and identify special cases? 8. EE. 7 a

Common Core Standard: 8. EE. ─ Analyze and solve linear equations and pairs of simultaneous linear equations. . 7. Solve linear equations in one variable. a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).

Objective - To solve equations with variables on both sides.

SPECIAL CASES Not all equations can be solved. Most of the time, you will find the equation has ONE SOLUTION. This is the most common situation. 3 x - 5 = 2 x + 12 -2 x x - 5 = 12 +5 +5 x = 17 One Solution

SPECIAL CASES Sometimes both sides of the equation are IDENTICAL to each other. i. e. 2 x = 2 x This is called an IDENTITY. Think about how many numbers you can substitute for x that will make the equation true. An IDENTITY has INFINITE solutions.

INFINITE SOLUTIONS When there an infinite number of solutions, we say the solution is x = all real numbers Steps 3 x + 8 = 2(x + 4) + x 3 x + 8 = 2 x + 8 + x 3 x + 8 = 3 x + 8 -3 x 8 = 8 True ! Reason Given Identity Infinite solutions x = all real numbers

SPECIAL CASES Sometimes the variables on both sides of the equation cancel each other out to make an equation that is false. i. e. 2 = 7 This is NOT TRUE. Nothing you can do will make the equation true. A NOT TRUE equation has NO SOLUTION.

NO SOLUTION When there are no solutions, we say the answer is no solution Steps 3 x + 2 = 2(x - 1) + x 3 x + 2 = 2 x - 2 + x 3 x + 2 = 3 x - 2 -3 x 2 = -2 False ! Not Identical Not True Reason Given No Solution

Solve: 4(y - 2) + 6 y = 7(y - 8) - 3(10 - y) 4 y - 8 + 6 y = 7 y - 56 - 30 + 3 y 10 y - 8 = 10 y - 86 -10 y -8 = -86 False Statement No Solution

Solve: 3(4 + k) - 2(3 k + 4) = 5(k - 3) - (8 k - 19) 12 + 3 k - 6 k - 8 = 5 k - 15 - 8 k + 19 -3 k + 4 = -3 k + 4 +3 k 4 = 4 True Statement Infinitely Many Solutions k = all real numbers

APPLICATION Use a variable equation to solve: Find a number that is 20 more than 4 times its opposite. Let x = the number -x = the opposite x = 4(-x) + 20 x = -4 x + 20 +4 x 5 x = 20 x=4