Bell Ringer Solve the following: 1. ) 7(4 – t) = -84 2 Bell Ringer Solve the following: 1.) 7(4 – t) = -84 2.) 78 = 3c +12 – c + 4 Go over homework from yesterday Pg 274 6-18 even

6-4 Solving Equations with the Variable on Both Sides Objectives: to solve equations with the variable on both sides. to solve equations containing grouping symbols.

To solve for a variable the variable must be positive with no number beside it!!  x = 10 Solved - x = -5 Not solved 2x = 5 Not solved -3x =6 Not solved Bottom page of notes

To solve these equations, Use the addition or subtraction property to move all terms from one side of the equal sign to another. 6x - 3 = 2x + 13 -2x -2x 4x - 3 = 13 Use the multiplication or division to bust a term up. (Separate the coefficient (number) from the variable (letter)) 4x = 16 4 4 x = 4

Let’s see a few examples: Make variable on 1 side of equation (do it so the variable is positive) 1) 6x - 3 = 2x + 13 -2x -2x 4x - 3 = 13 +3 +3 4x = 16 4 4 x = 4 Be sure to check your answer! 6(4) - 3 = 2(4) + 13 24 - 3 = 8 + 13 21 = 21

Let’s try another! Check: 3(1.5) + 1 = 7(1.5) - 5 4.5 + 1 = 10.5 - 5 5.5 = 5.5 2) 3n + 1 = 7n - 5 -3n -3n 1 = 4n - 5 +5 +5 6 = 4n 4 4 Reduce! 3 = n 2 Rewrite as n = 3/2

Here’s a tricky one! 3) 5 + 2(y + 4) = 5(y - 3) + 10 Distribute first. 5 + 2y + 8 = 5y - 15 + 10 Next, combine like terms. 2y + 13 = 5y - 5 Now solve. (Subtract 2y.) 13 = 3y - 5 (Add 5.) 18 = 3y (Divide by 3.) 6 = y Rewrite as y = 6 Check: 5 + 2(6 + 4) = 5(6 - 3) + 10 5 + 2(10) = 5(3) + 10 5 + 20 = 15 + 10 25 = 25

Let’s try one with fractions! Steps: Multiply each term by the least common denominator (8) to eliminate fractions. Solve for x. Add 2x. Add 6. Divide by 6. 3 - 2x = 4x - 6 3 = 6x - 6 9 = 6x so x = 3/2 2 * 1 * 2 * 4 *

(Notice there’s no variable at all in both cases.) Two special cases: 6(4 + y) - 3 = 4(y - 3) + 2y 24 + 6y - 3 = 4y - 12 + 2y 21 + 6y = 6y - 12 - 6y - 6y 21 = -12 Never true! 21 ≠ -12 Write ………. NO SOLUTION! 3(a + 1) - 5 = 3a - 2 3a + 3 - 5 = 3a - 2 3a - 2 = 3a - 2 -3a -3a -2 = -2 Always true! We write IDENTITY. (Notice there’s no variable at all in both cases.)

Try a few on your own: 9x + 7 = 3x - 5 8 - 2(y + 1) = -3y + 1 8 - 1 z = 1 z - 7 2 4

The answers: x = -2 y = -5 z = 20

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