Two Step Equations Test Review
Inverse Operations
Understand the concept of inverse operations which are used to isolate the variable in equations.
Operations and Inverses4
Addition (+) and **Subtraction (-)
Example: B + 6 = 5
Inverse operation: -6 (Both sides)
Result: B = -1
Subtraction (-) and Addition (+)
Example: -5 = C - 7
Inverse operation: +7 (Both sides)
Result: C = 2
Multiplication (×) and Division (÷)
Example: 7x = 77
Divide both sides by 7
Result: x = 11
Sample Problem Check
Check: 7x=77, Substitute x = 11
Check: 7(11) = 77
Verified: 77=77
Two-Step Equations
Definition: A two-step equation is one where at least two operations are involved to solve for a variable.
Examples
Example: -2x - 5 = -13
Add 5 to both sides: -2x = -8
Divide both sides by -2: x = 4
Check: -2(4) - 5 = -13
Result: -8 - 5 = -13
Verified: -13 = -13
Example: To get rid of a fractional coefficient
Equation: 3(x + 1) = -3/5
Divide both sides by 3: x + 1 = -1/5
Subtract 1 from both sides: x = -1 - 1/5 = -6/5
Check: 3(-6/5 + 1) = -3
Result: 3(-1/5) = -3
Verified: -3 = -3
Your Try:
Solve: 3x + 2 = 8
Subtract 2 from both sides: 3x = 6
Divide by 3: x = 2
Check: 3(2) + 2 = 8, Result: 6 + 2 = 8, Verified: 8 = 8
Example: Solve: -2(x + 4) = -14
Distribute -2: -2x - 8 = -14
Add 8 to both sides: -2x = -6
Divide by -2: x = 3
Story Problems
Use problem-solving skills to translate words into equations by identifying key phrases for operations:
Key Operations
Addition: Increased by, sum of, plus
Subtraction: Less than, decreased by, minus
Multiplication: Times, product of, of
Division: Divided by, quotient of, per
Example Problem 1:
Problem: Brandy wants to buy a digital camera that costs $300. Suppose she saves $15 each week. How many weeks until she can buy the camera?
Equation: 15w = 300
Solve: w = 300 / 15 = 20 weeks
Check: 15(20) = 300, Confirmed.
Example Problem 2:
Problem: Danny is three times and an additional 6 years older than Mira. If Danny is 21, how old is Mira?
Equation: 3m + 6 = 21
Solve: 3m = 21 - 6 = 15; m = 5 years old
Check: 3(5) + 6 = 21, Confirmed.
Example Problem 3:
Problem: The IMS basketball team wants to order sweatshirts to commemorate their amazing year. Each sweatshirt costs $12.50 to print with a $15 shipping and handling charge. If the team has $140 to spend, how many sweatshirts can they order?
Equation: 12.50s + 15 = 140
Solve: 12.50s = 140 - 15;
Divide both sides by 12.50 to find s.
Check: Calculate total costs after finding s to ensure it does not exceed $140.
Problem 1: 2(x - 5) = 10
Distribute 2: 2x - 10 = 10
Add 10 to both sides: 2x = 20
Divide by 2: x = 10
Problem 2: -3x + 4 = -2
Subtract 4 from both sides: -3x = -6
Divide by -3: x = 2
Problem 3: 5m - 7 = 3
Add 7 to both sides: 5m = 10
Divide by 5: m = 2
Problem 4: 4(y + 2) = 20
Divide by 4: y + 2 = 5
Subtract 2 from both sides: y = 3
Problem 5: -2z - 5 = -13
Add 5 to both sides: -2z = -8
Divide by -2: z = 4
Example Problem 5:
Problem: Sarah is saving to buy a bike that costs $250. She currently has $40 and saves $20 each week. How many weeks will it take her to save enough money for the bike?
Equation: 20w + 40 = 250
Solve: 20w = 250 - 40; w = 210 / 20 = 10 weeks
Check: 20(10) + 40 = 250, Confirmed.
Example Problem 6:
Problem: Mark is four years older than Anna. If Anna is currently 12 years old, how old is Mark?
Equation: m = a + 4, where a = 12.
Solve: m = 12 + 4; m = 16 years old
Check: 12 + 4 = 16, Confirmed.
Example Problem 7:
Problem: A school fundraiser sells tickets for $5 each. If they sell 150 tickets, how much money do they make?
Equation: 5t = 5(150)
Solve: t = 150; Total = 5(150) = $750
Check: 5 * 150 = 750, Confirmed.
Example Problem 8:
Problem: A tank holds 500 liters of water. If it is currently filled with 125 liters, how many liters need to be added to fill the tank?
Equation: x + 125 = 500
Solve: x = 500 - 125; x = 375 liters
Check: 375 + 125 = 500, Confirmed.