Solve MultiStep Equations Review of Chapter 2 Steps

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Solve Multi-Step Equations Review of Chapter 2

Solve Multi-Step Equations Review of Chapter 2

Steps to Solve Equations with Variables on Both Sides v 1) Simplify each side

Steps to Solve Equations with Variables on Both Sides v 1) Simplify each side v Get rid of double Negatives v Distribute v Combine Like Terms v 2) Move variables to same side v “Smaller to the bigger” v 3) Solve by using INVERSE Operations

Use Steps to Solve Equation: 2(x + 7) + 3 = 5 x -

Use Steps to Solve Equation: 2(x + 7) + 3 = 5 x - 1 2 x + 14 + 3 = 5 x – 1 2 x + 17 = 5 x - 1 -2 x 17 = 3 x - 1 +1 +1 18 = 3 x 3 3 x= 6 bkevil 1) distribute 2) Combine like terms 3) Get variables on same side – use inverse operation 4) Solve 2 step equation

Check - replace “x” with solution 2(x + 7) + 3 = 5 x

Check - replace “x” with solution 2(x + 7) + 3 = 5 x - 1 2(6 + 7) + 3 =5(6) - 1 2(13) + 3 =5(6) - 1 26 + 3 = 30 - 1 29 = 29 X=6 bkevil 1) Replace X = 6 2) Follow order of operations on both sides of equation 3) Checks is the solution to the equation

Use Steps to Solve Equation: -3 x + 4 = 5 x – 8

Use Steps to Solve Equation: -3 x + 4 = 5 x – 8 +3 x 4 = 8 x - 8 +8 +8 12 = 8 x 8 8 x = 3/2 Get variables on same side of equation – use inverse operation (add 3 x) Solve 2 step equation bkevil

Use Steps to Solve Equation: 4(1 – 2 x) = 4 – 6 x

Use Steps to Solve Equation: 4(1 – 2 x) = 4 – 6 x 4 – 8 x = 4 - 6 x +8 x + 8 x 4 = 4 + 2 x -4 -4 0 = 2 x 2 2 x= 0 bkevil Get rid of ( ) -- distribute Get variables on same side – use inverse operation (add 6 x) Solve 2 step equation Undo by using inverse -2 undo 2 nd +4 -4 undo 1 st

Use Steps to Solve Equation: 9 + 5 x = 5 x + 9

Use Steps to Solve Equation: 9 + 5 x = 5 x + 9 -5 x Get variables on same side of equation – use inverse operation (subtract 5 x) 9=9 When solving, if you get a TRUE STATEMENT, then that means that any real number works. Infinite Solutions bkevil

Use Steps to Solve Equation: 6 x – 1 = 6 x – 8

Use Steps to Solve Equation: 6 x – 1 = 6 x – 8 -6 x Get variables on same side of equation – use inverse operation (subtract 6 x) -1 = - 8 The variables zeroed out and remaining is a false statement where a number is equal to a different number, so there will be no number that will work in the equation. x = no solutions The solution is no real numbers or empty set bkevil

Review Steps to Solve Equations with Variables on Both Sides v 1) Simplify each

Review Steps to Solve Equations with Variables on Both Sides v 1) Simplify each side v Get rid of double Negatives v Distribute v Combine Like Terms v 2) Move variables to same side v “Smaller to the bigger” v 3) Solve by using INVERSE Operations bkevil

Solve the equation. 1. 2 m – 6 + 4 m = 12 ANSWER

Solve the equation. 1. 2 m – 6 + 4 m = 12 ANSWER 3 2. 6 a – 5(a – 1) = 11 ANSWER 6

Solve the equation. 3. A charter bus company charges $11. 25 per ticket plus

Solve the equation. 3. A charter bus company charges $11. 25 per ticket plus a handling charge of $. 50 per ticket, and a $15 fee for booking the bus. If a group pays $297 to charter a bus, how many tickets did they buy? ANSWER 24 tickets

Independent Practice Solve the equation. 1. 8 g – 2 + g = 16

Independent Practice Solve the equation. 1. 8 g – 2 + g = 16 ANSWER 2 2. 3 b + 2(b – 4) = 47 ANSWER 11 3. – 6 + 4(2 c + 1) = – 34 ANSWER – 4

Independent Practice 4. 2 (x – 6) = 12 3 ANSWER 24 5. Joe

Independent Practice 4. 2 (x – 6) = 12 3 ANSWER 24 5. Joe drove 405 miles in 7 hours. He drove at a rate of 55 miles per hour during the first part of the trip and 60 miles per hour during the second part. How many hours did he drive at a rate of 55 miles per hour? ANSWER 3 h

EXAMPLE 1 Solve an equation with variables on both sides Solve 7 – 8

EXAMPLE 1 Solve an equation with variables on both sides Solve 7 – 8 x = 4 x – 17 7 – 8 x + 8 x = 4 x – 17 + 8 x 7 = 12 x – 17 24 = 12 x 2=x Write original equation. Add 8 x to each side. Simplify each side. Add 17 to each side. Divide each side by 12. ANSWER The solution is 2. Check by substituting 2 for x in the original equation.

EXAMPLE 1 Solve an equation with variables on both sides CHECK 7 – 8

EXAMPLE 1 Solve an equation with variables on both sides CHECK 7 – 8 x = 4 x – 17 ? 7 – 8(2) = 4(2) – 17 ? Write original equation. Substitute 2 for x. – 9 = 4(2) – 17 Simplify left side. – 9 = – 9 Simplify right side. Solution checks.

EXAMPLE 2 Solve an equation with grouping symbols 1 Solve 9 x – 5

EXAMPLE 2 Solve an equation with grouping symbols 1 Solve 9 x – 5 = 4 (16 x + 60). 1 9 x – 5 = (16 x + 60) 4 Write original equation. 9 x – 5 = 4 x + 15 Distributive property 5 x – 5 = 15 Subtract 4 x from each side. 5 x = 20 x=4 Add 5 to each side. Divide each side by 5.

GUIDED PRACTICE for Examples 1 and 2 Solve the equation. Check your solution. 1.

GUIDED PRACTICE for Examples 1 and 2 Solve the equation. Check your solution. 1. 24 – 3 m = 5 m ANSWER 3

GUIDED PRACTICE for Examples 1 and 2 Solve the equation. Check your solution. 2.

GUIDED PRACTICE for Examples 1 and 2 Solve the equation. Check your solution. 2. 20 + c = 4 c – 7 ANSWER 9

GUIDED PRACTICE for Examples 1 and 2 Solve the equation. Check your solution. 3.

GUIDED PRACTICE for Examples 1 and 2 Solve the equation. Check your solution. 3. 9 – 3 k = 17 k – 2 k ANSWER – 8

GUIDED PRACTICE for Examples 1 and 2 Solve the equation. Check your solution. 4.

GUIDED PRACTICE for Examples 1 and 2 Solve the equation. Check your solution. 4. 5 z – 2 = 2(3 z – 4) ANSWER 6

GUIDED PRACTICE for Examples 1 and 2 Solve the equation. Check your solution. 5.

GUIDED PRACTICE for Examples 1 and 2 Solve the equation. Check your solution. 5. 3 – 4 a = 5(a – 3) ANSWER 2

GUIDED PRACTICE for Examples 1 and 2 Solve the equation. Check your solution. 6.

GUIDED PRACTICE for Examples 1 and 2 Solve the equation. Check your solution. 6. 2 8 y – 6 = 3 (6 y + 15) ANSWER 4

EXAMPLE 3 Solve a real-world problem CAR SALES A car dealership sold 78 new

EXAMPLE 3 Solve a real-world problem CAR SALES A car dealership sold 78 new cars and 67 used cars this year. The number of new cars sold by the dealership has been increasing by 6 cars each year. The number of used cars sold by the dealership has been decreasing by 4 cars each year. If these trends continue, in how many years will the number of new cars sold be twice the number of used cars sold?

EXAMPLE 3 Solve a real-world problem SOLUTION Let x represent the number of years

EXAMPLE 3 Solve a real-world problem SOLUTION Let x represent the number of years from now. So, 6 x represents the increase in the number of new cars sold over x years and – 4 x represents the decrease in the number of used cars sold over x years. Write a verbal model. 78 + 6 x =2( 67 + (– 4 x) )

EXAMPLE 3 Solve a real-world problem 78 + 6 x = 2(67 – 4

EXAMPLE 3 Solve a real-world problem 78 + 6 x = 2(67 – 4 x) Write equation. 78 + 6 x = 134 – 8 x Distributive property 78 + 14 x = 134 14 x = 56 x= 4 Add 8 x to each side. Subtract 78 from each side. Divide each side by 14. ANSWER The number of new cars sold will be twice the number of used cars sold in 4 years.

EXAMPLE 3 CHECK Solve a real-world problem You can use a table to check

EXAMPLE 3 CHECK Solve a real-world problem You can use a table to check your answer. YEAR Used car sold 0 67 1 63 2 59 3 55 4 51 New car sold 78 84 90 96 102

GUIDED PRACTICE 7. for Example 3 WHAT IF? In Example 3, suppose the car

GUIDED PRACTICE 7. for Example 3 WHAT IF? In Example 3, suppose the car dealership sold 50 new cars this year instead of 78. In how many years will the number of new cars sold be twice the number of used cars sold? ANSWER 6 yr

EXAMPLE 4 Identify the number of solutions of an equation Solve the equation, if

EXAMPLE 4 Identify the number of solutions of an equation Solve the equation, if possible. a. 3 x = 3(x + 4) b. 2 x + 10 = 2(x + 5) SOLUTION a. 3 x = 3(x + 4) 3 x = 3 x + 12 Original equation Distributive property The equation 3 x = 3 x + 12 is not true because the number 3 x cannot be equal to 12 more than itself. So, the equation has no solution. This can be demonstrated by continuing to solve the equation.

EXAMPLE 4 Identify the number of solutions of an equation 3 x – 3

EXAMPLE 4 Identify the number of solutions of an equation 3 x – 3 x = 3 x + 12 – 3 x 0 = 12 Subtract 3 x from each side. Simplify. ANSWER The statement 0 = 12 is not true, so the equation has no solution.

EXAMPLE 41 b. Identify the number of solutions of an equation 2 x +

EXAMPLE 41 b. Identify the number of solutions of an equation 2 x + 10 = 2(x + 5) Original equation 2 x + 10 = 2 x + 10 Distributive property ANSWER Notice that the statement 2 x + 10 = 2 x + 10 is true for all values of x. So, the equation is an identity, and the solution is all real numbers.

GUIDED PRACTICE for Example 4 Solve the equation, if possible. 8. 9 z +

GUIDED PRACTICE for Example 4 Solve the equation, if possible. 8. 9 z + 12 = 9(z + 3) ANSWER no solution

GUIDED PRACTICE for Example 4 Solve the equation, if possible. 9. 7 w +

GUIDED PRACTICE for Example 4 Solve the equation, if possible. 9. 7 w + 1 = 8 w + 1 ANSWER 0

GUIDED PRACTICE for Example 4 Solve the equation, if possible. 10. 3(2 a +

GUIDED PRACTICE for Example 4 Solve the equation, if possible. 10. 3(2 a + 2) = 2(3 a + 3) ANSWER identity