Solved Exercises on Solving Multi-Step Equations in Grade 7

Master solving multi-step equations: combining like terms, distributive property, variables on both sides, and real-world applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Combining Like Terms
Exercise 1
Solve: 3x + 2x + 8 = 23
Definition:

Multi-step equation: An equation that requires more than two operations to solve.

Multi-step equation method:
  1. Combine like terms on the same side
  2. Undo addition or subtraction
  3. Undo multiplication or division
  4. Check the solution
Original Equation
3x + 2x + 8 = 23
Step 1: Combine like terms
5x + 8 = 23
Step 2: Subtract 8 from both sides
5x = 15
Step 3: Divide both sides by 5
x = 3
Step 1: Combine like terms

On the left side, combine 3x and 2x:

3x + 2x = 5x

So: 5x + 8 = 23

Step 2: Undo addition

Subtract 8 from both sides to eliminate the constant term:

5x + 8 - 8 = 23 - 8

5x = 15

Step 3: Undo multiplication

Divide both sides by 5 to isolate x:

5x ÷ 5 = 15 ÷ 5

x = 3

Step 4: Check the solution

Substitute x = 3 back into the original equation:

3(3) + 2(3) + 8 = 9 + 6 + 8 = 23 ✓

x = 3
Final answer:

x = 3

Applied rules:

• Combining like terms: Add coefficients of terms with same variable

• Order of operations reversal: Undo in reverse order of operations

• Balancing: Perform the same operation on both sides

2 Distributive Property
Exercise 2
Solve: 2(x + 5) + 3 = 17
Definition:

Distributive property: a(b + c) = ab + ac. Apply this first when solving equations with parentheses.

Original Equation
2(x + 5) + 3 = 17
Step 1: Apply distributive property
2x + 10 + 3 = 17
Step 2: Combine constants
2x + 13 = 17
Step 3: Solve two-step equation
x = 2
Step 1: Apply distributive property

Multiply 2 by both terms inside the parentheses:

2(x + 5) = 2x + 10

So: 2x + 10 + 3 = 17

Step 2: Combine constants

On the left side, combine 10 and 3:

2x + 13 = 17

Step 3: Undo addition

Subtract 13 from both sides:

2x = 4

Step 4: Undo multiplication

Divide both sides by 2:

x = 2

Step 5: Check the solution

Substitute x = 2 back into the original equation:

2(2 + 5) + 3 = 2(7) + 3 = 14 + 3 = 17 ✓

x = 2
Final answer:

x = 2

Applied rules:

• Distributive property: Multiply the outside factor by each term inside

• Order of operations: Distribute before combining like terms

• Systematic approach: Work step by step to avoid errors

3 Variables on Both Sides
Exercise 3
Solve: 4x + 3 = 2x + 9
Definition:

Variables on both sides: Move all variables to one side and all constants to the other side.

Original Equation
4x + 3 = 2x + 9
Step 1: Subtract 2x from both sides
2x + 3 = 9
Step 2: Subtract 3 from both sides
2x = 6
Step 3: Divide both sides by 2
x = 3
Step 1: Move variables to one side

Subtract 2x from both sides to get all x terms on the left:

4x - 2x + 3 = 2x - 2x + 9

2x + 3 = 9

Step 2: Move constants to other side

Subtract 3 from both sides to get constants on the right:

2x + 3 - 3 = 9 - 3

2x = 6

Step 3: Isolate the variable

Divide both sides by 2:

x = 3

Step 4: Check the solution

Substitute x = 3 back into the original equation:

Left side: 4(3) + 3 = 12 + 3 = 15

Right side: 2(3) + 9 = 6 + 9 = 15 ✓

x = 3
Final answer:

x = 3

Applied rules:

• Variable collection: Move all variable terms to one side

• Constant collection: Move all constant terms to the other side

• Balancing: Perform same operation on both sides

Multi-Step Equations Rules and Methods
a(x + b) + c = d \Rightarrow x = \frac{d - c}{a} - b
Distributive Property Equation
Combining Like Terms
ax + bx + c = d → (a + b)x + c = d
Combine coefficients first
Distributive Property
a(x + b) + c = d → ax + ab + c = d
Distribute first, then solve
Variables on Both Sides
ax + b = cx + d → (a - c)x = d - b
Move variables to one side
Key definitions:

Multi-step equation: An algebraic equation requiring more than two operations to solve for the variable.

Like terms: Terms with identical variables raised to identical powers that can be combined.

Distributive property: The rule that a(b + c) = ab + ac, used to eliminate parentheses.

Variables on both sides: Equations where the variable appears on both sides of the equals sign.

Systematic solving: Following a logical sequence of operations to isolate the variable.

Solving methodology:
  1. Simplify each side: Distribute and combine like terms if necessary
  2. Move variables to one side: Add or subtract to collect all variable terms on one side
  3. Move constants to other side: Add or subtract to collect all constants on the other side
  4. Isolate the variable: Use multiplication or division to get the variable alone
  5. Verify solution: Substitute back into original equation
Tip 1: Always simplify each side first by distributing and combining like terms.
Tip 2: Move the variable term with the smaller coefficient to avoid negative coefficients.
Tip 3: Keep the equation balanced by doing the same thing to both sides.
Tip 4: Always check your solution by substituting it back into the original equation.
Common errors: Forgetting to distribute, combining unlike terms, not collecting all variables on one side, making sign errors, not verifying the solution.
Exam preparation: Practice all types of multi-step equations, master distributive property, work with negative numbers, develop systematic solving approaches.
Essential rules:

• Distributive property: a(b + c) = ab + ac

• Like terms: Combine only terms with identical variable parts

• Variable collection: Move all variable terms to one side

• Balance principle: Whatever you do to one side, do to the other

• Verification: Always check your solution in the original equation

Solution: Exercises 4 to 5
4 Complex Distribution
Exercise 4
Solve: 3(x - 2) + 4(2x + 1) = 25
Definition:

Complex distribution: Multiple sets of parentheses require applying the distributive property to each set.

Original Equation
3(x - 2) + 4(2x + 1) = 25
Step 1: Distribute both terms
3x - 6 + 8x + 4 = 25
Step 2: Combine like terms
11x - 2 = 25
Step 3: Solve
x = 3
Step 1: Apply distributive property to both sets of parentheses

3(x - 2) = 3x - 6

4(2x + 1) = 8x + 4

So: 3x - 6 + 8x + 4 = 25

Step 2: Combine like terms

Combine variable terms: 3x + 8x = 11x

Combine constant terms: -6 + 4 = -2

So: 11x - 2 = 25

Step 3: Undo subtraction

Add 2 to both sides:

11x = 27

Step 4: Undo multiplication

Divide both sides by 11:

x = 27/11

Step 5: Check the solution

Substitute x = 27/11 back into the original equation:

3(27/11 - 2) + 4(2(27/11) + 1) = 3(27/11 - 22/11) + 4(54/11 + 11/11) = 3(5/11) + 4(65/11) = 15/11 + 260/11 = 275/11 = 25 ✓

x = 27/11
Final answer:

x = 27/11

Applied rules:

• Multiple distributions: Apply distributive property to each set of parentheses

• Systematic approach: Work through each step carefully

• Verification: Check solutions even when they're fractions

5 Real-World Application
Exercise 5
A company charges $20 per day plus a $15 service fee for equipment rental. Another company charges $25 per day with no service fee. For how many days would the costs be equal?
Definition:

Real-world applications: Setting up equations with variables on both sides to solve practical problems.

Problem Setup
Let x = number of days
Write Equation
20x + 15 = 25x
Solve
x = 3
Step 1: Define the variable

Let x = the number of days (unknown)

Step 2: Set up the equation

Company 1 cost: 20x + 15 (daily rate × days + service fee)

Company 2 cost: 25x (daily rate × days)

For equal costs: 20x + 15 = 25x

Step 3: Move variables to one side

Subtract 20x from both sides: 15 = 25x - 20x

15 = 5x

Step 4: Isolate the variable

Divide both sides by 5: x = 3

Step 5: Interpret the solution

The costs would be equal after 3 days

Step 6: Verify the solution

Company 1: 20(3) + 15 = 60 + 15 = $75

Company 2: 25(3) = $75 ✓

x = 3 (3 days)
Final answer:

The costs would be equal after 3 days

Applied rules:

• Problem translation: Convert real-world situations into mathematical equations

• Variable definition: Clearly define what the variable represents

• Equation setup: Set up equations based on equal conditions

Detailed Multi-Step Equations Guide
a(x + b) + c = d \Rightarrow x = \frac{d - c}{a} - b
Distributive Property Solution
Key definitions:

Multi-step equation: An algebraic equation that requires more than two operations to solve for the variable. These equations may involve combining like terms, using the distributive property, and having variables on both sides.

Like terms: Terms that have identical variables raised to identical powers. For example, 3x and 5x are like terms, but 3x and 3x² are not.

Distributive property: The mathematical rule that states a(b + c) = ab + ac. This is used to eliminate parentheses in equations.

Variables on both sides: Equations where the variable appears on both sides of the equals sign, requiring collection of all variable terms on one side.

Systematic solving: Following a logical sequence of operations to isolate the variable: simplify, collect terms, isolate, verify.

Complete solving methodology:
  1. Simplify each side: Apply distributive property and combine like terms
  2. Collect variables: Move all variable terms to one side of the equation
  3. Collect constants: Move all constant terms to the other side of the equation
  4. Isolate the variable: Use inverse operations to get the variable alone
  5. Verify solution: Substitute the answer back into the original equation
Tip 1: Always simplify each side completely before moving terms between sides.
Tip 2: When moving variables, move the term with the smaller coefficient to avoid negative coefficients.
Tip 3: Keep the equation balanced by performing the same operation on both sides.
Tip 4: Always check your solution by substituting it back into the original equation.
Common errors: Forgetting to distribute to all terms inside parentheses, combining unlike terms, making sign errors when moving terms, not collecting all variables on one side, failing to verify solutions, arithmetic mistakes in multi-step processes.
Applications: Comparing pricing plans, calculating break-even points, solving mixture problems, determining time-distance-rate relationships, financial planning, business decisions, and preparing for more advanced algebra.
Essential solving rules:

• Distributive property: a(b + c) = ab + ac

• Like terms: Only combine terms with identical variable parts

• Variable collection: Move all variable terms to one side

• Constant collection: Move all constant terms to the other side

• Balance principle: Whatever you do to one side, do to the other

Multi-Step Equations Guide

📊
Combining Like Terms

3x + 2x + 5 = 20

→ 5x + 5 = 20

→ 5x = 15

→ x = 3

Distributive Property

2(x + 3) + 4 = 12

→ 2x + 6 + 4 = 12

→ 2x + 10 = 12

→ x = 1

Variables on Both Sides

4x + 2 = 2x + 8

→ 2x + 2 = 8

→ 2x = 6

→ x = 3

Solution Verification

Always substitute

back into original

equation to verify

Questions & Answers

Question: When I have variables on both sides like 3x + 5 = 2x + 8, how do I know which side to move the variables to?

Answer: You can move variables to either side, but it's generally better to move them to the side with the larger coefficient to avoid negative coefficients.

For 3x + 5 = 2x + 8:

Option 1 (move to left side):

  • Subtract 2x from both sides: 3x - 2x + 5 = 8
  • Result: x + 5 = 8
  • Then: x = 3

Option 2 (move to right side):

  • Subtract 3x from both sides: 5 = 2x - 3x + 8
  • Result: 5 = -x + 8
  • Then: -3 = -x, so x = 3

Option 1 is easier because it avoids the negative coefficient! Move variables to the side with the larger coefficient.

Question: What do I do when I have parentheses like 2(x + 3) + 4? Should I combine like terms first?

Answer: No, you must apply the distributive property FIRST before combining like terms! Follow the order of operations in reverse.

For 2(x + 3) + 4:

  1. Step 1: Distribute the 2: 2(x + 3) = 2x + 6
  2. Step 2: Now you have: 2x + 6 + 4
  3. Step 3: Combine like terms: 2x + 10

Why this order matters:

  • You can't combine 6 and 4 while they're separated by parentheses
  • Distribution eliminates the parentheses
  • After distribution, you can identify all like terms

Remember: Distribute first, then combine like terms!

Question: How do I check my answer when the solution is a fraction like x = 3/4?

Answer: Substitute the fractional answer back into the ORIGINAL equation and simplify both sides to see if they're equal!

Example: If you solved 4x + 2 = 5 and got x = 3/4

Check by substituting:

  • Left side: 4(3/4) + 2 = 3 + 2 = 5
  • Right side: 5
  • Since 5 = 5, your answer is correct!

Key points for fractions:

  • When multiplying: (numerator × whole number) ÷ denominator
  • Be careful with signs when substituting negative fractions
  • Simplify completely to verify equality

Checking works the same way regardless of whether your answer is a whole number, fraction, or decimal!