Solving Equations With Variables on Both Sides of the equal sign

Introduction We’ve already worked on solving – 1-step equations: x – 3 = 4 – 2-step equations: 2x + 8 = -10 – Multi-step equations: 5(a – 3) + a = 21 What happens if there is a variable on both sides of the equal sign though? 3x + 2 = 4x - 1

Goal: Get the variable on one side of the equal (=) sign then solve. To do this we have to move the smaller variable. How do we move a variable? Apply the inverse operation to the whole term! EXAMPLE 3x + 2 = 4x x 2 = x = x Important Notes Apply operations directly below like terms! Bring equal sign straight down

y = 1 TRY THE FOLLOWING 8y – 9 = -3y + 23 – 4x = 18 + x +3y 11y – 9 = y = x 3 = x = 5x = x

Multi-Step Example 3(x + 4) = 2(x - 1) 3x + 12 = 2x - 2 Distributive Property -2x x + 12 = x = - 14 Then Solve!

Special Case #1 2x + 5 = 2x x 5 = -3 But is this possible? NO!!! So the answer is NO SOLUTIONS

Special Case #2 3(x + 1) - 5 = 3x - 2 3x + 3 – 5 = 3x - 2 3x - 2 = 3x – 2 -3x -3x -2 = -2 This is always true! Infinitely many solutions

Your Favorite and Mine…… WORD PROBLEMS Mr. Szabo needed to rent a moving van. Company A charges a rate of $90 per hour plus a $50 truck fee. Company B charges a $70 per hour plus fee plus a $90 truck fee. For what numbers of hours of rental is the cost for the two companies the same?

Define our variables first! Let h = the number of hours of rental Company ACompany B $90 per hour plus $50$70 per hour plus $90 = 90h + 50 = 70h h 20h + 50 = h = h = 2

Word Problem Answer The rental cost for the two companies is the same after 2 hours of rental. HoursCompany ACompany B h90h h $50$90 1$140$160 2$230 3$320$300

LET’S PRACTICE