Solved Exercises on Number of Solutions to Equations in Integrated Math 1

Master the three types of solutions: unique solution, no solution, infinite solutions through these 5 detailed exercises with comprehensive solutions.

Solution: Exercises 1 to 3
1 Unique solution
Exercise 1
Solve for x:
\(2x + 3 = 7\)
Definition:

Unique solution: An equation that has exactly one value for the variable that makes the equation true

Linear equation: An equation that can be written in the form \(ax + b = c\) where \(a \neq 0\)

Result: A single value for the variable

Unique solution method:

For equations with unique solutions:

  1. Isolate the variable: Use inverse operations to get the variable alone
  2. Verify: Substitute the solution back into the original equation
  3. Result: The variable equals a specific number
Original
\(2x + 3 = 7\)
Subtract 3
\(2x = 4\)
Divide by 2
\(x = 2\)
Step 1: Write the equation

\(2x + 3 = 7\)

Step 2: Undo addition by subtracting 3 from both sides

\(2x + 3 - 3 = 7 - 3\)

\(2x = 4\)

Step 3: Undo multiplication by dividing both sides by 2

\(\frac{2x}{2} = \frac{4}{2}\)

\(x = 2\)

Step 4: Verify the solution

Substitute \(x = 2\) into original: \(2(2) + 3 = 4 + 3 = 7\) ✓

\(x = 2\)
Final answer:

The equation has one unique solution: \(x = 2\)

Applied rules:

• Addition/Subtraction Property: Adding or subtracting the same number from both sides maintains equality

• Multiplication/Division Property: Dividing both sides by the same non-zero number maintains equality

• Unique Solution: When solving leads to \(x = \text{number}\), there is one solution

Tip: When you get \(x = \text{number}\), you have a unique solution!
Tip: Always verify your solution by substituting back into the original equation.
2 No solution
Exercise 2
Solve for x:
\(2x + 5 = 2x + 8\)
Definition:

No solution: An equation that has no value for the variable that makes the equation true

Contradiction: A statement that is always false, such as \(5 = 8\)

Result: An impossible statement after solving

Original
\(2x + 5 = 2x + 8\)
Subtract 2x
\(5 = 8\)
Step 1: Write the equation

\(2x + 5 = 2x + 8\)

Step 2: Move variables to both sides by subtracting 2x from both sides

\(2x + 5 - 2x = 2x + 8 - 2x\)

\(5 = 8\)

Step 3: Analyze the result

Since \(5 \neq 8\), this is a contradiction

Step 4: Conclusion

The equation has no solution because it leads to an impossible statement

No solution
Final answer:

The equation has no solution because it leads to the contradiction \(5 = 8\)

Applied rules:

• Variable Elimination: When variables cancel out and constants are unequal, there is no solution

• Contradiction Recognition: If solving leads to a false statement like \(a = b\) where \(a \neq b\), there is no solution

• No Solution: The equation represents parallel lines that never intersect

Tip: When variables cancel out and you get a false statement like \(5 = 8\), there is no solution!
Tip: No solution occurs when the coefficients of the variable are the same but the constants are different.
3 Infinite solutions
Exercise 3
Solve for x:
\(3x + 6 = 3(x + 2)\)
Definition:

Infinite solutions: An equation that is true for any value of the variable

Identity: A statement that is always true, such as \(6 = 6\)

Result: A statement that is always true after solving

Original
\(3x + 6 = 3(x + 2)\)
Distribute
\(3x + 6 = 3x + 6\)
Subtract 3x
\(6 = 6\)
Step 1: Write the equation

\(3x + 6 = 3(x + 2)\)

Step 2: Distribute on the right side

\(3(x + 2) = 3x + 6\), so: \(3x + 6 = 3x + 6\)

Step 3: Move variables to both sides by subtracting 3x from both sides

\(3x + 6 - 3x = 3x + 6 - 3x\)

\(6 = 6\)

Step 4: Analyze the result

Since \(6 = 6\) is always true, any value of x will satisfy the equation

Step 5: Conclusion

The equation has infinitely many solutions because it leads to a true statement

Infinite solutions
Final answer:

The equation has infinitely many solutions because it leads to the identity \(6 = 6\)

Applied rules:

• Distributive Property: Multiply the outside term by each term inside parentheses

• Identity Recognition: If solving leads to a true statement like \(a = a\), there are infinite solutions

• Infinite Solutions: The equation represents the same line, so all points satisfy it

Tip: When variables cancel out and you get a true statement like \(6 = 6\), there are infinitely many solutions!
Tip: Infinite solutions occur when both sides of the equation are identical after simplification.
Types of Solutions and Recognition Methods
\(\text{Unique: } ax + b = c \Rightarrow x = \frac{c-b}{a} \text{ (when } a \neq 0)\)
Unique Solution
\(\text{No Solution: } ax + b = ax + c \text{ (when } b \neq c)\)
No Solution
\(\text{Infinite: } ax + b = ax + b \text{ (always true)}\)
Infinite Solutions
Unique
\(x = \text{number}\)
Exactly one solution exists
No Solution
\(a = b \text{ (false)}\)
Variables cancel, constants differ
Infinite
\(a = a \text{ (true)}\)
Both sides are identical
Equality Property: Both sides of an equation must remain equal when performing operations.
Solution Types: Every linear equation has either one, none, or infinitely many solutions.
Additional Examples: Mixed Types
4 No solution with distribution
Exercise 4
Solve for x:
\(2(x + 3) = 2x + 9\)
Definition:

No solution with distribution: An equation requiring distribution that leads to a contradiction

Key concept: After distributing and simplifying, variables may cancel while constants remain unequal

Original
\(2(x + 3) = 2x + 9\)
Distribute
\(2x + 6 = 2x + 9\)
Subtract 2x
\(6 = 9\)
Step 1: Write the equation

\(2(x + 3) = 2x + 9\)

Step 2: Distribute on the left side

\(2(x + 3) = 2x + 6\), so: \(2x + 6 = 2x + 9\)

Step 3: Move variables to both sides by subtracting 2x from both sides

\(2x + 6 - 2x = 2x + 9 - 2x\)

\(6 = 9\)

Step 4: Analyze the result

Since \(6 \neq 9\), this is a contradiction

Step 5: Conclusion

The equation has no solution because it leads to an impossible statement

No solution
Final answer:

The equation has no solution because it leads to the contradiction \(6 = 9\)

Applied rules:

• Distributive Property: Multiply the outside term by each term inside parentheses

• Variable Elimination: When variables cancel out and constants are unequal, there is no solution

• Contradiction Recognition: If solving leads to a false statement like \(a = b\) where \(a \neq b\), there is no solution

Tip: Always distribute completely before determining the number of solutions!
5 Infinite solutions with distribution
Exercise 5
Solve for x:
\(4(x - 1) + 2 = 4x - 2\)
Definition:

Infinite solutions with distribution: An equation requiring distribution that leads to an identity

Key concept: After distributing and simplifying, both sides become identical

Original
\(4(x - 1) + 2 = 4x - 2\)
Distribute
\(4x - 4 + 2 = 4x - 2\)
Simplify
\(4x - 2 = 4x - 2\)
Subtract 4x
\(-2 = -2\)
Step 1: Write the equation

\(4(x - 1) + 2 = 4x - 2\)

Step 2: Distribute on the left side

\(4(x - 1) = 4x - 4\), so: \(4x - 4 + 2 = 4x - 2\)

Step 3: Simplify the left side

\(4x - 4 + 2 = 4x - 2\), so: \(4x - 2 = 4x - 2\)

Step 4: Move variables to both sides by subtracting 4x from both sides

\(4x - 2 - 4x = 4x - 2 - 4x\)

\(-2 = -2\)

Step 5: Analyze the result

Since \(-2 = -2\) is always true, any value of x will satisfy the equation

Step 6: Conclusion

The equation has infinitely many solutions because it leads to a true statement

Infinite solutions
Final answer:

The equation has infinitely many solutions because it leads to the identity \(-2 = -2\)

Applied rules:

• Distributive Property: Multiply the outside term by each term inside parentheses

• Identity Recognition: If solving leads to a true statement like \(a = a\), there are infinite solutions

• Infinite Solutions: The equation represents the same line, so all points satisfy it

Tip: When both sides of the equation are identical after simplification, there are infinitely many solutions!
Comprehensive Guide to Number of Solutions
\(\text{Every linear equation has: } \begin{cases} \text{One solution} & \text{if } a \neq 0 \\ \text{No solution} & \text{if } a = 0, b \neq c \\ \text{Infinitely many} & \text{if } a = 0, b = c \end{cases}\)
Solution Classification
Key definitions:

Unique solution: An equation that has exactly one value for the variable that makes the equation true

No solution: An equation that has no value for the variable that makes the equation true

Infinite solutions: An equation that is true for any value of the variable

Contradiction: A statement that is always false

Identity: A statement that is always true

Complete methodology:
  1. Solve the equation: Use inverse operations to isolate the variable
  2. Observe the result: Look at what remains after simplification
  3. Classify the solution: Based on the final statement
  4. Verify: Check your classification makes sense
Tip 1: If you get \(x = \text{number}\), you have a unique solution!
Tip 2: If variables cancel out and you get a false statement like \(5 = 8\), there is no solution.
Tip 3: If variables cancel out and you get a true statement like \(6 = 6\), there are infinitely many solutions.
Tip 4: Always distribute completely before determining the number of solutions.
Tip 5: Graphically, unique solution = intersecting lines, no solution = parallel lines, infinite = same line.
Common errors: Misidentifying true vs. false statements, not distributing completely, confusing no solution with infinite solutions, making calculation errors.
Exam preparation: Practice recognizing all three types, focus on distribution before classifying, master verification techniques.
Recognition Rules:

• Unique Solution: \(x = \text{number}\) or equivalent form

• No Solution: Variables cancel, constants differ: \(a = b\) where \(a \neq b\)

• Infinite Solutions: Both sides identical: \(a = a\) or equivalent

• Verification: Test with specific values to confirm classification

Number of Solutions Classification

📊
Solution Recognition Process
1
Solve
2
Simplify
3
Analyze
4
Classify
Solution Types Summary
Unique: x = number → one solution
No Solution: false statement → no solution
Infinite: true statement → infinite solutions
Graph: intersecting, parallel, same line
Master All Three Solution Types to Excel in Algebra!

Questions & Answers

Question: How can I tell the difference between no solution and infinite solutions? They both seem similar when variables cancel out.

Answer: Great question! The key difference is in the constants that remain after variables cancel:

No Solution: After variables cancel, you get a false statement with different constants:

  • Example: \(2x + 5 = 2x + 8\) → subtract 2x → \(5 = 8\) (FALSE!)
  • Result: No solution

Infinite Solutions: After variables cancel, you get a true statement with identical constants:

  • Example: \(3x + 6 = 3x + 6\) → subtract 3x → \(6 = 6\) (TRUE!)
  • Result: Infinite solutions

Remember: False statement = no solution, True statement = infinite solutions!

Question: Why do some equations have no solution? It seems like there should always be an answer.

Answer: Think of equations geometrically as lines on a graph:

Unique solution: Two lines that intersect at exactly one point (different slopes)

No solution: Two parallel lines that never intersect (same slope, different y-intercepts)

Infinite solutions: Two lines that are exactly the same (same slope and y-intercept)

For example, \(2x + 5 = 2x + 8\) represents two parallel lines:

  • Line 1: \(y = 2x + 5\)
  • Line 2: \(y = 2x + 8\)

These lines have the same slope (2) but different y-intercepts (5 vs 8), so they're parallel and never meet - hence no solution!

Question: How do I verify that an equation has infinitely many solutions?

Answer: There are two ways to verify infinite solutions:

  1. Algebraic verification: After solving, you should get a true statement like \(6 = 6\) or \(0 = 0\)
  2. Substitution verification: Try substituting different values for the variable - they should all work!

For example, with \(3x + 6 = 3(x + 2)\), which simplifies to \(6 = 6\):

  • Try \(x = 0\): \(3(0) + 6 = 6\) and \(3(0 + 2) = 6\) → both equal 6 ✓
  • Try \(x = 1\): \(3(1) + 6 = 9\) and \(3(1 + 2) = 9\) → both equal 9 ✓
  • Try \(x = -5\): \(3(-5) + 6 = -9\) and \(3(-5 + 2) = -9\) → both equal -9 ✓

Any value of x works, confirming infinitely many solutions!