Solved Exercises on Solving Systems by Substitution in Grade 8

Master solving systems by substitution: variable isolation, substitution technique, and solution verification through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic Substitution
Exercise 1
Solve the system by substitution: y = 2x + 3 and 3x + y = 18.
Definition:

Substitution Method: A technique to solve systems by replacing one variable with an equivalent expression

Substitution method steps:
  1. Isolate one variable in one equation
  2. Substitute the isolated variable's expression into the other equation
  3. Solve for the remaining variable
  4. Substitute back to find the other variable
  5. Verify the solution in both original equations
Step 1
y = 2x + 3
Step 2
3x + (2x + 3) = 18
Step 3
x = 3, y = 9
Step 1: Identify the isolated variable

The first equation already has y isolated: y = 2x + 3

Step 2: Substitute into the second equation

Replace y in the second equation with (2x + 3):

3x + (2x + 3) = 18

Step 3: Solve for x

3x + 2x + 3 = 18

5x + 3 = 18

5x = 15

x = 3

Step 4: Find y by substitution

Substitute x = 3 into y = 2x + 3:

y = 2(3) + 3 = 6 + 3 = 9

Step 5: Verify the solution

Check in first equation: y = 2(3) + 3 = 6 + 3 = 9 ✓

Check in second equation: 3(3) + 9 = 9 + 9 = 18 ✓

Solution: (3, 9)
Final answer:

The solution to the system is x = 3 and y = 9, or the ordered pair (3, 9).

Applied rules:

• Substitution: Replace variable with equivalent expression

• Algebraic Manipulation: Solve linear equations step by step

• Verification: Check solution in both original equations

2 Variable Isolation
Exercise 2
Solve the system by substitution: x + 2y = 11 and 3x - y = 5.
Definition:

Variable Isolation: Solving for one variable in terms of the other to enable substitution

Step 1
x = 11 - 2y
Step 2
3(11 - 2y) - y = 5
Step 3
y = 4, x = 3
Step 1: Isolate a variable

From the first equation x + 2y = 11, solve for x:

x = 11 - 2y

Step 2: Substitute into the other equation

Replace x in the second equation with (11 - 2y):

3(11 - 2y) - y = 5

Step 3: Solve for y

33 - 6y - y = 5

33 - 7y = 5

-7y = -28

y = 4

Step 4: Find x by substitution

Substitute y = 4 into x = 11 - 2y:

x = 11 - 2(4) = 11 - 8 = 3

Step 5: Verify the solution

Check in first equation: 3 + 2(4) = 3 + 8 = 11 ✓

Check in second equation: 3(3) - 4 = 9 - 4 = 5 ✓

Solution: (3, 4)
Final answer:

The solution to the system is x = 3 and y = 4, or the ordered pair (3, 4).

Applied rules:

• Variable Isolation: Solve for easiest variable first

• Substitution: Replace variable with equivalent expression

• Verification: Check solution in both original equations

3 Fractional Coefficients
Exercise 3
Solve the system by substitution: y = (1/2)x + 3 and 2x + y = 13.
Definition:

Fractional Coefficients: Variables with fractional multipliers that require careful arithmetic handling

Step 1
y = (1/2)x + 3
Step 2
2x + (1/2)x + 3 = 13
Step 3
x = 4, y = 5
Step 1: Identify the isolated variable

The first equation already has y isolated: y = (1/2)x + 3

Step 2: Substitute into the second equation

Replace y in the second equation with ((1/2)x + 3):

2x + ((1/2)x + 3) = 13

Step 3: Solve for x

2x + (1/2)x + 3 = 13

(4/2)x + (1/2)x = 10

(5/2)x = 10

x = 10 × (2/5) = 4

Step 4: Find y by substitution

Substitute x = 4 into y = (1/2)x + 3:

y = (1/2)(4) + 3 = 2 + 3 = 5

Step 5: Verify the solution

Check in first equation: y = (1/2)(4) + 3 = 2 + 3 = 5 ✓

Check in second equation: 2(4) + 5 = 8 + 5 = 13 ✓

Solution: (4, 5)
Final answer:

The solution to the system is x = 4 and y = 5, or the ordered pair (4, 5).

Applied rules:

• Fraction Arithmetic: Careful handling of fractional coefficients

• Substitution: Replace variable with equivalent expression

• Verification: Check solution in both original equations

Substitution Method Rules and Procedures
\(\begin{cases} y = mx + b \\ ax + cy = d \end{cases} \Rightarrow a x + c(mx + b) = d\)
Substitution Process
Isolate
\(y = \text{expression}\)
Solve for one variable
Substitute
\(\text{variable} = \text{equivalent}\)
Replace in other equation
Solve
\(\text{single variable equation}\)
Find remaining variable
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

Substitution Method: A technique that replaces one variable with an equivalent expression to solve systems

Variable Isolation: Solving for one variable in terms of others to enable substitution

Equivalent Expression: An expression that has the same value as the original variable

Verification: Checking that the solution satisfies all original equations

Complete substitution methodology:
  1. Choose variable: Select the variable that's easiest to isolate
  2. Isolate variable: Solve one equation for the chosen variable
  3. Substitute: Replace the isolated variable in the other equation
  4. Solve: Solve the resulting single-variable equation
  5. Back-substitute: Use found value to solve for the other variable
  6. Verify: Check solution in both original equations
Tip 1: Choose the variable with coefficient 1 or -1 for easier isolation.
Tip 2: Be careful with signs when substituting negative expressions.
Tip 3: Always distribute coefficients when substituting expressions with multiple terms.
Tip 4: Always verify your solution by substituting back into both original equations.

Common errors: Sign errors when distributing negatives, arithmetic mistakes, forgetting to verify solutions, incorrect distribution.
Real-world applications: Economics, physics, chemistry, business optimization, and resource allocation problems.
Essential substitution principles:

• Variable Isolation: Solve for the simplest variable first

• Substitution: Replace variable with equivalent expression

• Distribution: Distribute coefficients when substituting expressions

• Verification: Always check solution in original equations

• Arithmetic: Careful handling of fractions and decimals

Solution: Exercises 4 to 5
4 Real-World Application
Exercise 4
A store sells apples for $2 each and oranges for $3 each. If someone bought 8 fruits for $20, how many of each did they buy?
Definition:

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

Define Variables
x = apples, y = oranges
Set Up System
x + y = 8, 2x + 3y = 20
Solution
x = 4, y = 4
Step 1: Define variables

Let x = number of apples

Let y = number of oranges

Step 2: Set up the system of equations

Equation 1 (total fruits): x + y = 8

Equation 2 (total cost): 2x + 3y = 20

Step 3: Isolate a variable

From equation 1: x = 8 - y

Step 4: Substitute into the other equation

Replace x in equation 2 with (8 - y):

2(8 - y) + 3y = 20

Step 5: Solve for y

16 - 2y + 3y = 20

16 + y = 20

y = 4

Step 6: Find x

Substitute y = 4 into x = 8 - y:

x = 8 - 4 = 4

Step 7: Verify the solution

Check in first equation: 4 + 4 = 8 ✓

Check in second equation: 2(4) + 3(4) = 8 + 12 = 20 ✓

4 apples, 4 oranges
Final answer:

The person bought 4 apples and 4 oranges.

Applied rules:

• Word Problem Modeling: Translate scenario into equations

• Variable Definition: Assign variables to unknowns

• Substitution Method: Solve system using substitution

5 Age Problem
Exercise 5
The sum of two people's ages is 30. One person is 4 years older than the other. 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 = 30, x = y + 4
Solution
x = 17, y = 13
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 = 30

Equation 2 (age difference): x = y + 4

Step 3: Notice one variable is already isolated

Equation 2 already has x isolated: x = y + 4

Step 4: Substitute into the other equation

Replace x in equation 1 with (y + 4):

(y + 4) + y = 30

Step 5: Solve for y

y + 4 + y = 30

2y + 4 = 30

2y = 26

y = 13

Step 6: Find x

Substitute y = 13 into x = y + 4:

x = 13 + 4 = 17

Step 7: Verify the solution

Check in first equation: 17 + 13 = 30 ✓

Check in second equation: 17 = 13 + 4 ✓

Ages: 17 and 13
Final answer:

The two people are 17 and 13 years old.

Applied rules:

• Word Problem Modeling: Translate scenario into equations

• Variable Definition: Assign variables to unknowns

• Substitution Method: Solve system using substitution

Detailed Summary: Solving Systems by Substitution
\(\begin{cases} ax + by = c \\ dx + ey = f \end{cases} \Rightarrow \text{Solve one for variable, substitute into other}\)
Substitution 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

Substitution Method: A technique that involves solving one equation for one variable and substituting that expression into the other equation

Variable Isolation: The process of solving an equation for a specific variable in terms of the others

Equivalent Expression: An expression that has the same value as the original variable, allowing for substitution

Back-Substitution: Using a found value to determine the remaining variable(s) in the system

Verification: The process of checking that the solution satisfies all original equations in the system

Complete substitution methodology:
  1. Examine the system: Look for an equation where a variable is already isolated or easy to isolate
  2. Isolate a variable: Solve one equation for one variable in terms of the others
  3. Substitute: Replace the isolated variable in the other equation with its equivalent expression
  4. Solve: Solve the resulting single-variable equation
  5. Back-substitute: Use the found value to determine the remaining variable(s)
  6. Verify: Check that the solution satisfies both original equations
  7. Interpret: Understand the meaning of the solution in the context of the problem
Tip 1: Always choose the variable with the simplest coefficient (preferably 1 or -1) for isolation.
Tip 2: When substituting expressions with multiple terms, use parentheses to ensure correct distribution.
Tip 3: Be especially careful with signs when substituting negative expressions.
Tip 4: Always verify your solution by substituting back into both original equations.

Common misconceptions: Thinking substitution only works when a variable is already isolated, forgetting to distribute coefficients, not checking solutions in both equations, making arithmetic errors with negative numbers.
Memorization aids: "ISOLATE, SUBSTITUTE, SOLVE, CHECK", "Solve one, plug into the other", "Find one variable, find the other".
Critical substitution principles:

• Variable Choice: Select the easiest variable to isolate first

• Substitution: Replace variable with equivalent expression

• Distribution: Carefully distribute coefficients when substituting

• Verification: Always check solution in original equations

• Arithmetic: Pay attention to signs and fraction operations

• System Consistency: Verify solution satisfies all equations

Visualizing Substitution Method: Before and After
Exercise 6: Substitution Visualization
Visual demonstration of the substitution process:
Original system: y = 2x + 1 and 3x + y = 11
After substitution: 3x + (2x + 1) = 11 → 5x + 1 = 11
Result: Single variable equation leading to solution

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

  • Original system: Two equations with two variables
  • Substitution: Replaces one variable with an expression
  • Result: Single equation with one variable
  • Solution: Back-substitution finds the other variable

Questions & Answers

Question: How do I decide which variable to isolate when using substitution?

Answer: Here's how to choose which variable to isolate:

  • Look for coefficients of 1 or -1: Variables with these coefficients are easiest to isolate
  • Check for already isolated variables: If one equation already has a variable by itself, use that
  • Consider arithmetic complexity: Choose the variable that avoids fractions or decimals when possible
  • Check for simplicity: Pick the variable that requires the fewest steps to isolate

Example: In the system {x + 2y = 7, 3x - y = 5}, it's easier to isolate x from the first equation (x = 7 - 2y) than to isolate y from either equation.

The goal is to minimize complexity in the subsequent substitution step.

Remember, you can always solve for either variable, but choosing wisely makes the process easier.

Question: What should I do when my substitution results in fractions?

Answer: When fractions appear in substitution problems:

  • Keep working: Fractions are normal in mathematics and don't mean you made an error
  • Be careful with arithmetic: Pay attention to adding/subtracting fractions and multiplying/dividing
  • Find common denominators: When adding or subtracting fractions
  • Verify your answer: Fractional solutions are valid if they check out in both equations

Example: If you get x = 3/2, substitute back into both equations to verify:

When checking, convert to improper fractions or decimals to make arithmetic easier.

Fractions in solutions are completely acceptable as long as they satisfy both original equations.

Always double-check your fraction arithmetic, as this is where most errors occur.

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

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 {x + y = 7, 2x - y = 1} with solution (2, 5):

  • First equation: 2 + 5 = 7 ✓
  • Second equation: 2(2) - 5 = 4 - 5 = -1 ≠ 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.