Linear Equations: Two-Step

Two- Step Equations are equations which have two operations to undo.

Ex: – 9 – 14m = 3 + 4 = 10

Non Ex: y + 4 = -3

Which is an example of a two-step equation? -5x – 6 < -8 OR 7y – 9 = 3

2x

The goal is to get the variable by itself.

To undo an operation, perform the inverse (opposite) of that operation.

The goal of solving a two-step equation is __________________________________

Operation Inverse Operation

Steps:

1. Circle variable/ Line down equal sign

2. Undo addition or subtraction.

3. Undo multiplication or division.

4. Check answer.

Ex 1: 10 = + 2

Steps:1. Circle variable/ Line down equal sign2. Undo addition or subtraction.3. Undo multiplication or division.4. Check answer.

4

m

Ex 2: -b + 6 = -11

Steps:1. Circle variable/ Line down equal sign2. Undo addition or subtraction.3. Undo multiplication or division.4. Check answer.

Ex 3: 1 = + 5

Steps:1. Circle variable/ Line down equal sign2. Undo addition or subtraction.3. Undo multiplication or division.4. Check answer.

12

k

Ex 4: 8 – y = 14

Steps:1. Circle variable/ Line down equal sign2. Undo addition or subtraction.3. Undo multiplication or division.4. Check answer.

Ex 5: – 5 = 2

Steps:1. Circle variable/ Line down equal sign2. Undo addition or subtraction.3. Undo multiplication or division.4. Check answer.

9

x

Ex 6: 4 = – c + 11

Steps:1. Circle variable/ Line down equal sign2. Undo addition or subtraction.3. Undo multiplication or division.4. Check answer.

Linear Equations: 2-Step Word Problems

The first step in solving any word problem is to identify the unknown.

Usually, this is the question at the end of the problem.

The unknown is represented by a variable.

What does a variable represent?

The cost of renting a jet ski is $60 plus $12 per hour. How many hours can the jet ski be used if the total cost is $144?

X = # of _ (unknown)

The second step is to write an equation.

In a 2-step word problem, the standard form is always the same.

____________x + ________ = ____________(multiple amount) (one-time amount) (total amount)

What is the standard form of a 2-step word problem?

The cost of renting a jet ski is $60 plus $12 per hour. How many hours can the jet

ski be used if the total cost is $144?

Steps:

1. Identify the unknown.

2. Write the standard form.

3. Plug in the amount values.

4. Solve the equation.

5. Check the answer/Is it reasonable?

Ex 1: The cost of renting a car is $75 plus $25 per day. The total cost of Damon’s bill is $300. How many days did he rent the car?

Steps:1. Identify the unknown.2. Write the standard form.3. Plug in the amount values.4. Solve the equation.5. Check the answer/Is it reasonable?

Ex 2: The cost to rent a plane is $40 plus $250 per hour of use for the pilot. How many hours can the plane be used, if the cost to rent the plane is $1900?

Steps:1. Identify the unknown.2. Write the standard form.3. Plug in the amount values.4. Solve the equation.5. Check the answer/Is it reasonable?

Ex 3: Caleb’s total bill for renting a motorcycle is $300. The cost of renting the motorcycle is $180 plus $24 per day. How many days did Caleb rent the motorcycle?

Steps:1. Identify the unknown.2. Write the standard form.3. Plug in the amount values.4. Solve the equation.5. Check the answer/Is it reasonable?

Linear Equations/Inequalities: Variables on Both Sides

Ex: 2x – 5 = 8x + 7 x – 5 < 2 + 3x

Non Ex: x + 4 = – 2

Give an example of an equation with variables on both sides.

To solve an equation with variables on both sides, get the variables on the same side of the equation.

Add or subtract one of the variable terms from both sides of the equation.

2x – 5 = 8x + 7 -2x -2x (subtract 2x from both sides) – 5 = 6x + 7

T/F?: Get the variables on opposite sides of the equation.What do you add or subtract from both sides of the equation?

Steps:

1. Distribute.

2. Collect like terms.

3. Get variables on same side of equation.

4. Undo addition or subtraction.

5. Undo multiplication or division.

6. Check.

Ex 1: 4x + 7 > 1 + 5x

Steps:1. Distribute.2. Collect like terms.3. Get variables on same side of equation.4. Undo addition or subtraction.5. Undo multiplication or division.6. Check.

Ex 2: 3x + 7 = 5x + 9

Steps:1. Distribute.2. Collect like terms.3. Get variables on same side of equation.4. Undo addition or subtraction.5. Undo multiplication or division.6. Check.

Ex 3: 6x + 3 < 8x – 21

Steps:1. Distribute.2. Collect like terms.3. Get variables on same side of equation.4. Undo addition or subtraction.5. Undo multiplication or division.6. Check.

Linear Equations/Inequalites: Multi-Step (Collecting Like Terms)

A multi-step equation/inequality takes at least 3 steps to solve.

Ex: 2c + c + 12 = 78 12 > 3m + 8 – 5m – 1

Non Ex: 6y – 5 = -17

Which is an example of a multi-step with collecting like terms?

5x + 6 = 4 – 2x OR 9 = 7x – 1 + 4x – 2

Steps:

1. Distribute.

2. Collect like terms.

3. Get variables on same side of equation.

4. Undo addition or subtraction.

5. Undo multiplication or division.

6. Check.

Ex 1: 78 > 4c + 9 – 7c + 3

Steps:1. Distribute.2. Collect like terms.3. Get variables on same side of equation.4. Undo addition or subtraction.5. Undo multiplication or division.6. Check.

Ex 2: -23 = 2a – 1 – 8a + 8

Steps:1. Distribute.2. Collect like terms.3. Get variables on same side of equation.4. Undo addition or subtraction.5. Undo multiplication or division.6. Check.

Ex 3: -4x + 18 – 4x – 2 < -40

Steps:1. Distribute.2. Collect like terms.3. Get variables on same side of equation.4. Undo addition or subtraction.5. Undo multiplication or division.6. Check.

Linear Equations/Inequalities: Multi-Step (CLT/VBS)

Ex: x + 5 = x + 9 – x 10x + 3 – 3x > 9 – 5x – 16

Non Ex: 6x – 5 = -17

Give an example of a multi-step with collecting like terms and variables on both sides.

Steps:

1. Distribute.

2. Collect like terms.

3. Get variables on same side of equation.

4. Undo addition or subtraction.

5. Undo multiplication or division.

6. Check.

Ex 1: 21 + 6x = 30 + 7x – 7

Steps:1. Distribute.2. Collect like terms.3. Get variables on same side of equation.4. Undo addition or subtraction.5. Undo multiplication or division.6. Check.

Ex 2: 10 + 3x – 3 > -5x – 7 + 10x

Steps:1. Distribute.2. Collect like terms.3. Get variables on same side of equation.4. Undo addition or subtraction.5. Undo multiplication or division.6. Check.

Ex 3: 5 – 2x > 3x – 7x + 25

Steps:1. Distribute.2. Collect like terms.3. Get variables on same side of equation.4. Undo addition or subtraction.5. Undo multiplication or division.6. Check.

Linear Equations/Inequalities: Multi-Step (DP/CLT/VBS)

Ex: -2(x – 4) + x < 12 + 5x 3y – 21 + y = 3(y + 8) – 5

Non Ex: 3x + 6 – 2x = -4

Give an example of a multi-step using the DP, CLT, and VBS.

Steps:

1. Distribute.

2. Collect like terms.

3. Get variables on same side of equation.

4. Undo addition or subtraction.

5. Undo multiplication or division.

6. Check.

Ex 1: -27 = -6 + 3(y + 8)

Steps:1. Distribute.2. Collect like terms.3. Get variables on same side of equation.4. Undo addition or subtraction.5. Undo multiplication or division.6. Check.

Ex 2: -2(m – 4) + 5 < 17

Steps:1. Distribute.2. Collect like terms.3. Get variables on same side of equation.4. Undo addition or subtraction.5. Undo multiplication or division.6. Check.

Ex 3: 38 = -7x – 3(x + 4)

Steps:1. Distribute.2. Collect like terms.3. Get variables on same side of equation.4. Undo addition or subtraction.5. Undo multiplication or division.6. Check.