3-4 Solving Equations with Variables on Both Sides Extension of AF4.1 Solve two-step linear equations in one variable over the rational numbers. Also covered: AF1.1 California Standards

3-4 Solving Equations with Variables on Both Sides Solve. 4x + 6 = x Additional Example 1A: Solving Equations with Variables on Both Sides – 4x 6 = –3x To collect the variable terms on one side, subtract 4x from both sides. Since x is multiplied by -3, divide both sides by –3. –2 = x 6 –3 –3x –3 = Notes X = -2

3-4 Solving Equations with Variables on Both Sides You can always check your solution by substituting the value back into the original equation. Helpful Hint

3-4 Solving Equations with Variables on Both Sides Solve. 9b – 6 = 5b + 18 Additional Example 1B: Solving Equations with Variables on Both Sides - Notes – 5b 4b – 6 = 18 4b4b = To collect the variable terms on one side, subtract 5b from both sides. Since b is multiplied by 4, divide both sides by 4. b = b = 24 Since 6 is subtracted from 4b, add 6 to both sides.

3-4 Solving Equations with Variables on Both Sides if the variables in an equation are eliminated and the resulting statement is false, the equation has no solution. Helpful Hint Notes

3-4 Solving Equations with Variables on Both Sides Solve. 5x + 8 = x Check It Out! Example 1A – 5x Since x is multiplied by –4, divide both sides by –4. –2 = x 8 –4 –4x –4 = To collect the variable terms on one side, subtract 5x from both sides. X = -2 Always have the variable on the left side of the equation.

3-4 Solving Equations with Variables on Both Sides Solve. 3b – 2 = 2b + 12 – 2b b – 2 = b = 14 Since 2 is subtracted from b, add 2 to both sides. Check It Out! Example 1B To collect the variable terms on one side, subtract 2b from both sides.

3-4 Solving Equations with Variables on Both Sides To solve more complicated equations, you may need to first simplify by combining like terms or clearing fractions. Then add or subtract to collect variable terms on one side of the equation. Finally, use properties of equality to isolate the variable.

3-4 Solving Equations with Variables on Both Sides Solve. 10z – 15 – 4z = 8 – 2z – 15 Additional Example 2A: Solving Multi-Step Equations with Variables on Both Sides z – 15 = –2z – 7Combine like terms. + 2z Add 2z to both sides. 8z – 15 = – 7 8z = 8 z = 1 Add 15 to both sides. Divide both sides by 8. 8z = Notes

3-4 Solving Equations with Variables on Both Sides Additional Example 2B: Solving Multi-Step Equations with Variables on Both Sides Multiply by the LCD, 20. 4y + 12y – 15 = 20y – 14 16y – 15 = 20y – 14Combine like terms. y5y y53y – = y – 20 ( ) + 20 ( ) – 20 ( ) = 20(y) – 20 ( ) y5y5 3y53y Notes

3-4 Solving Equations with Variables on Both Sides Additional Example 2B Continued Add 14 to both sides. –15 = 4y – 14 –1 = 4y + 14 –1 4 4y4y 4 = Divide both sides by 4. –1 4 = y 16y – 15 = 20y – 14 – 16y Subtract 16y from both sides.

3-4 Solving Equations with Variables on Both Sides Solve. 12z – 12 – 4z = 6 – 2z + 32 Check It Out! Example 2A 12z – 12 – 4z = 6 – 2z z – 12 = –2z + 38Combine like terms. + 2z Add 2z to both sides. 10z – 12 = 38 10z = 50 z = 5 Add 12 to both sides. Divide both sides by z = Elbow Partners

3-4 Solving Equations with Variables on Both Sides Multiply by the LCD, 24. 6y + 20y + 18 = 24y – 18 26y + 18 = 24y – 18Combine like terms. y4y y65y = y – y4y y65y ( ) = 24 ( ) y4y y65y y – 24 ( ) + 24 ( ) + 24 ( ) = 24(y) – 24 ( ) y4y4 5y65y Check It Out! Example 2B Face Partners

3-4 Solving Equations with Variables on Both Sides Subtract 18 from both sides. 2y + 18 = – 18 2y = –36 – 18 –36 2 2y2y 2 = Divide both sides by 2. y = –18 26y + 18 = 24y – 18 – 24y Subtract 24y from both sides. Check It Out! Example 2B Continued

3-4 Solving Equations with Variables on Both Sides Additional Example 3: Business Application Daisy’s Flowers sells a rose bouquet for $39.95 plus $2.95 for every rose. A competing florist sells a similar bouquet for $26.00 plus $4.50 for every rose. Find the number of roses that would make both florists' bouquets cost the same price. What is the price? Daisy’s: c = r Write an equation for each service. Let c represent the total cost and r represent the number of roses. total cost is flat fee plus cost for each rose Other: c = r

3-4 Solving Equations with Variables on Both Sides Additional Example 3 Continued r = r Now write an equation showing that the costs are equal. – 2.95r = r Subtract 2.95r from both sides. – Subtract from both sides = 1.55r r 1.55 = Divide both sides by = r The two bouquets from either florist would cost the same when purchasing 9 roses.

3-4 Solving Equations with Variables on Both Sides Additional Example 3 Continued To find the cost, substitute 9 for r into either equation. Daisy’s: The cost for a bouquet with 9 roses at either florist is $ c = r c = (9) c = c = 66.5 Other florist: c = r c = (9) c = c = 66.5