Solved Exercises on Solving Systems by Elimination in Grade 8

Master solving systems by elimination: adding/subtracting equations, coefficient manipulation, and solution verification through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Addition Elimination
Exercise 1
Solve the system by elimination: 2x + 3y = 13 and 4x - 3y = 5.
Definition:

Elimination Method: A technique to solve systems by adding or subtracting equations to eliminate one variable

Elimination method steps:
  1. Arrange equations with like terms aligned
  2. Identify which variable can be eliminated by adding or subtracting
  3. Add or subtract the equations to eliminate the variable
  4. Solve for the remaining variable
  5. Substitute back to find the other variable
  6. Verify the solution in both original equations
Step 1
2x + 3y = 13
Step 2
4x - 3y = 5
Step 3
6x = 18 → x = 3
Step 1: Align the equations

2x + 3y = 13

4x - 3y = 5

Step 2: Identify elimination opportunity

The coefficients of y are +3 and -3 (opposites)

Adding the equations will eliminate y

Step 3: Add the equations

(2x + 3y) + (4x - 3y) = 13 + 5

2x + 3y + 4x - 3y = 18

6x = 18

Step 4: Solve for x

6x = 18

x = 3

Step 5: Find y by substitution

Substitute x = 3 into first equation:

2(3) + 3y = 13

6 + 3y = 13

3y = 7

y = 7/3

Step 6: Verify the solution

Check in first equation: 2(3) + 3(7/3) = 6 + 7 = 13 ✓

Check in second equation: 4(3) - 3(7/3) = 12 - 7 = 5 ✓

Solution: (3, 7/3)
Final answer:

The solution to the system is x = 3 and y = 7/3, or the ordered pair (3, 7/3).

Applied rules:

• Elimination: Add equations when coefficients are opposites

• Variable Elimination: Opposite coefficients cancel out

• Verification: Check solution in both original equations

2 Subtraction Elimination
Exercise 2
Solve the system by elimination: 3x + 2y = 16 and 3x - y = 7.
Definition:

Subtraction Elimination: A technique where subtracting equations eliminates a variable with identical coefficients

Step 1
3x + 2y = 16
Step 2
3x - y = 7
Step 3
3y = 9 → y = 3
Step 1: Align the equations

3x + 2y = 16

3x - y = 7

Step 2: Identify elimination opportunity

The coefficients of x are both +3 (identical)

Subtracting the equations will eliminate x

Step 3: Subtract the equations

(3x + 2y) - (3x - y) = 16 - 7

3x + 2y - 3x + y = 9

3y = 9

Step 4: Solve for y

3y = 9

y = 3

Step 5: Find x by substitution

Substitute y = 3 into second equation:

3x - 3 = 7

3x = 10

x = 10/3

Step 6: Verify the solution

Check in first equation: 3(10/3) + 2(3) = 10 + 6 = 16 ✓

Check in second equation: 3(10/3) - 3 = 10 - 3 = 7 ✓

Solution: (10/3, 3)
Final answer:

The solution to the system is x = 10/3 and y = 3, or the ordered pair (10/3, 3).

Applied rules:

• Elimination: Subtract equations when coefficients are identical

• Variable Elimination: Identical coefficients cancel out when subtracted

• Verification: Check solution in both original equations

3 Coefficient Manipulation
Exercise 3
Solve the system by elimination: 2x + 3y = 8 and 5x + 2y = 9.
Definition:

Coefficient Manipulation: Multiplying equations by constants to create elimination opportunities

Step 1
Multiply first by 2, second by 3
Step 2
4x + 6y = 16, 15x + 6y = 27
Step 3
x = 1, y = 2
Step 1: Examine the coefficients

First equation: 2x + 3y = 8

Second equation: 5x + 2y = 9

No coefficients are the same or opposites

Step 2: Manipulate coefficients to create elimination

To eliminate y, multiply first equation by 2 and second by 3:

First: 2(2x + 3y) = 2(8) → 4x + 6y = 16

Second: 3(5x + 2y) = 3(9) → 15x + 6y = 27

Step 3: Subtract equations to eliminate y

(15x + 6y) - (4x + 6y) = 27 - 16

15x + 6y - 4x - 6y = 11

11x = 11

x = 1

Step 4: Find y by substitution

Substitute x = 1 into original first equation:

2(1) + 3y = 8

2 + 3y = 8

3y = 6

y = 2

Step 5: Verify the solution

Check in first equation: 2(1) + 3(2) = 2 + 6 = 8 ✓

Check in second equation: 5(1) + 2(2) = 5 + 4 = 9 ✓

Solution: (1, 2)
Final answer:

The solution to the system is x = 1 and y = 2, or the ordered pair (1, 2).

Applied rules:

• Coefficient Manipulation: Multiply equations to create elimination opportunities

• Least Common Multiple: Use LCM to determine multiplication factors

• Verification: Check solution in both original equations

Elimination Method Rules and Procedures
\(\begin{cases} ax + by = c \\ dx + ey = f \end{cases} \Rightarrow \text{Multiply and add/subtract to eliminate}\)
Elimination Process
Addition
\(a = -d \Rightarrow \text{add equations}\)
Opposite coefficients
Subtraction
\(a = d \Rightarrow \text{subtract equations}\)
Identical coefficients
Manipulation
\(\text{Multiply by constants}\)
Create elimination opportunity
Key definitions:

System of Equations: Two or more equations with the same variables that must be satisfied simultaneously

Solution to System: An ordered pair (or set of values) that satisfies all equations in the system

Elimination Method: A technique that adds or subtracts equations to eliminate one variable

Coefficient Manipulation: Multiplying equations by constants to create elimination opportunities

Variable Elimination: Removing one variable to solve for the other

Verification: Checking that the solution satisfies all original equations

Complete elimination methodology:
  1. Align equations: Arrange like terms in columns
  2. Examine coefficients: Look for same or opposite coefficients
  3. Manipulate if needed: Multiply equations to create elimination opportunities
  4. Add/subtract: Combine equations to eliminate one variable
  5. Solve: Find the value of the remaining variable
  6. Substitute: Use found value to solve for the other variable
  7. Verify: Check solution in both original equations
Tip 1: Look for opposite coefficients first (they eliminate with addition).
Tip 2: Look for identical coefficients next (they eliminate with subtraction).
Tip 3: When multiplying equations, multiply both sides by the same constant.
Tip 4: Always verify your solution by substituting back into both original equations.

Common errors: Sign errors when subtracting equations, arithmetic mistakes, forgetting to multiply both sides of equation, not verifying solutions.
Real-world applications: Economics, physics, chemistry, business optimization, and resource allocation problems.
Essential elimination principles:

• Addition: Use when coefficients are opposites

• Subtraction: Use when coefficients are identical

• Manipulation: Multiply equations to create elimination opportunities

• Verification: Always check solution in original equations

• Arithmetic: Careful handling of signs and operations

Solution: Exercises 4 to 5
4 Real-World Application
Exercise 4
A store sells pencils for $2 each and erasers for $3 each. If someone bought 10 items for $24, how many of each did they buy?
Definition:

Word Problem Modeling: Translating real-world scenarios into mathematical equations for solution

Define Variables
x = pencils, y = erasers
Set Up System
x + y = 10, 2x + 3y = 24
Solution
x = 6, y = 4
Step 1: Define variables

Let x = number of pencils

Let y = number of erasers

Step 2: Set up the system of equations

Equation 1 (total items): x + y = 10

Equation 2 (total cost): 2x + 3y = 24

Step 3: Manipulate coefficients for elimination

Multiply equation 1 by 2 to eliminate x when subtracting:

2(x + y) = 2(10) → 2x + 2y = 20

Step 4: Subtract equations to eliminate x

(2x + 3y) - (2x + 2y) = 24 - 20

2x + 3y - 2x - 2y = 4

y = 4

Step 5: Find x by substitution

Substitute y = 4 into equation 1:

x + 4 = 10

x = 6

Step 6: Verify the solution

Check in first equation: 6 + 4 = 10 ✓

Check in second equation: 2(6) + 3(4) = 12 + 12 = 24 ✓

6 pencils, 4 erasers
Final answer:

The person bought 6 pencils and 4 erasers.

Applied rules:

• Word Problem Modeling: Translate scenario into equations

• Variable Definition: Assign variables to unknowns

• Elimination Method: Solve system using elimination

5 Age Problem
Exercise 5
The sum of two people's ages is 40. The difference between their ages is 8. Find their ages.
Definition:

Age Problems: Word problems involving relationships between people's ages, requiring system setup

Define Variables
x = age of first person, y = age of second
Set Up System
x + y = 40, x - y = 8
Solution
x = 24, y = 16
Step 1: Define variables

Let x = age of first person

Let y = age of second person

Step 2: Set up the system of equations

Equation 1 (sum of ages): x + y = 40

Equation 2 (age difference): x - y = 8

Step 3: Notice elimination opportunity

The coefficients of y are +1 and -1 (opposites)

Adding equations will eliminate y

Step 4: Add equations to eliminate y

(x + y) + (x - y) = 40 + 8

x + y + x - y = 48

2x = 48

x = 24

Step 5: Find y by substitution

Substitute x = 24 into equation 1:

24 + y = 40

y = 16

Step 6: Verify the solution

Check in first equation: 24 + 16 = 40 ✓

Check in second equation: 24 - 16 = 8 ✓

Ages: 24 and 16
Final answer:

The two people are 24 and 16 years old.

Applied rules:

• Word Problem Modeling: Translate scenario into equations

• Variable Definition: Assign variables to unknowns

• Elimination Method: Solve system using elimination

Detailed Summary: Solving Systems by Elimination
\(\begin{cases} ax + by = c \\ dx + ey = f \end{cases} \Rightarrow \text{Multiply and combine to eliminate variables}\)
Elimination Method Overview
Comprehensive definitions:

System of Linear Equations: A collection of linear equations with the same set of variables that must be satisfied simultaneously

Solution to a System: An ordered set of values that makes every equation in the system true when substituted

Elimination Method: A technique that involves adding or subtracting equations to eliminate one variable and solve for the other

Coefficient Manipulation: Multiplying equations by constants to create opportunities for elimination

Variable Elimination: The process of removing one variable from the system to solve for the remaining variable

Opposite Coefficients: When coefficients are the same number but opposite signs (+a and -a)

Identical Coefficients: When coefficients are the same number and same sign (+a and +a)

Complete elimination methodology:
  1. Align equations: Arrange like terms in columns for easy combination
  2. Examine coefficients: Look for opposite or identical coefficients
  3. Decide operation: Add for opposite coefficients, subtract for identical
  4. Manipulate if needed: Multiply equations to create elimination opportunities
  5. Combine equations: Add or subtract to eliminate one variable
  6. Solve: Find the value of the remaining variable
  7. Back-substitute: Use found value to determine the other variable
  8. Verify: Check solution in both original equations
Tip 1: Always look for opposite coefficients first (they eliminate with addition).
Tip 2: When multiplying equations, multiply both sides by the same constant to maintain equality.
Tip 3: Be careful with signs when subtracting equations (distribute the negative).
Tip 4: Always verify your solution by substituting back into both original equations.

Common misconceptions: Thinking elimination only works when coefficients are already the same/opposite, forgetting to multiply both sides of an equation, making sign errors when subtracting equations, not checking solutions in both equations.
Memorization aids: "OPPOSITES ADD, IDENTICALS SUBTRACT", "ELIMINATE, SOLVE, SUBSTITUTE, CHECK", "MULTIPLY TO CREATE ELIMINATION OPPORTUNITIES".
Critical elimination principles:

• Addition: Use when coefficients are opposites (+a and -a)

• Subtraction: Use when coefficients are identical (+a and +a)

• Manipulation: Multiply equations to create elimination opportunities

• Verification: Always check solution in original equations

• Arithmetic: Pay attention to signs and operations

• System Consistency: Solution must satisfy all equations

Visualizing Elimination Method: Before and After
Exercise 6: Elimination Visualization
Visual demonstration of the elimination process:
Original system: 2x + 3y = 13 and 4x - 3y = 5
After elimination: Adding equations gives 6x = 18
Result: Single variable equation leading to solution

Analysis: The chart visually demonstrates how the elimination method transforms a system of equations into a single equation.

  • Original system: Two equations with two variables
  • Elimination: Combines equations to remove one variable
  • Result: Single equation with one variable
  • Solution: Back-substitution finds the other variable

Questions & Answers

Question: How do I decide whether to add or subtract equations in elimination?

Answer: Here's how to decide:

  • Add equations: When coefficients of the variable you want to eliminate are OPPOSITES (like +3 and -3)
  • Subtract equations: When coefficients of the variable you want to eliminate are IDENTICAL (like +3 and +3)

Example 1: For 2x + 3y = 7 and 5x - 3y = 11, add equations to eliminate y (since +3 and -3 are opposites).

Example 2: For 4x + 2y = 10 and 4x - y = 3, subtract equations to eliminate x (since both have coefficient +4).

The goal is to make the coefficients of one variable sum to zero (for addition) or difference to zero (for subtraction).

Always verify your result by checking that the eliminated variable actually disappears from the combined equation.

Question: What should I do when none of the coefficients are the same or opposites?

Answer: When coefficients aren't already the same or opposites, you need to manipulate them:

  • Find LCM: Find the least common multiple of the coefficients you want to eliminate
  • Multiply: Multiply each equation by a constant to make coefficients equal
  • Proceed: Then use addition or subtraction to eliminate the variable

Example: For 3x + 2y = 8 and 5x + 3y = 13, to eliminate x:

  • Multiply first equation by 5: 15x + 10y = 40
  • Multiply second equation by 3: 15x + 9y = 39
  • Now subtract to eliminate x: (15x + 10y) - (15x + 9y) = 40 - 39

Remember to multiply both sides of each equation by the same constant to maintain equality.

You can eliminate either variable; choose the one that's easier to work with.

Question: How do I know if my solution is correct after using elimination?

Answer: Always verify your solution by substituting back into BOTH original equations:

  1. Take your solution: If you found x = 2, y = 5, substitute these values
  2. Check first equation: Replace variables in the first original equation
  3. Check second equation: Replace variables in the second original equation
  4. Verify equality: Both sides of each equation must be equal

Example: For system {2x + 3y = 16, x - y = 1} with solution (2, 4):

  • First equation: 2(2) + 3(4) = 4 + 12 = 16 ✓
  • Second equation: 2 - 4 = -2 ≠ 1 ❌

If your solution doesn't satisfy both equations, you made an error somewhere.

Verification is crucial because it catches calculation errors and confirms your answer is correct.

Always substitute into the original equations, not the modified ones created during elimination.