Solved Exercises on Systems with No Solution or Infinitely Many Solutions in Integrated Math 1

Master special cases of systems: inconsistent systems (no solution) and dependent systems (infinitely many solutions) through these 5 detailed exercises with comprehensive solutions.

Solution: Exercises 1 to 3
1 No Solution (Parallel Lines)
Exercise 1
Solve the system of equations:
\(2x + 3y = 6\) and \(2x + 3y = 12\)
Show that there is no solution and explain why.
Definition:

Inconsistent System: A system with no solutions; the equations represent parallel lines that never intersect

Method for identifying no solution:
  1. Try to solve the system using elimination or substitution
  2. If you arrive at a contradiction (like 0 = 6), the system has no solution
  3. Check that the lines have the same slope but different y-intercepts
  4. Confirm that the equations are parallel and distinct
System
2x + 3y = 6, 2x + 3y = 12
Elimination
0 = 6
Conclusion
No solution
Step 1: Align the equations

\(2x + 3y = 6\)

\(2x + 3y = 12\)

Step 2: Attempt to eliminate one variable

Subtract the first equation from the second:

\((2x + 3y) - (2x + 3y) = 12 - 6\)

\(0 = 6\)

Step 3: Analyze the result

The statement \(0 = 6\) is false, which is a contradiction

This means there is no ordered pair \((x, y)\) that satisfies both equations

Step 4: Verify by examining slopes

First equation: \(2x + 3y = 6\) → \(y = -\frac{2}{3}x + 2\) (slope = \(-\frac{2}{3}\))

Second equation: \(2x + 3y = 12\) → \(y = -\frac{2}{3}x + 4\) (slope = \(-\frac{2}{3}\))

Same slope, different y-intercepts → parallel lines

Step 5: Conclusion

Since the lines are parallel and distinct, they never intersect

The system has no solution

No solution
Final answer:

The system has no solution because the equations represent parallel lines with the same slope but different y-intercepts.

Applied rules:

• Contradiction: False statements like 0 = 6 indicate no solution

• Parallel Lines: Same slope, different y-intercept → No intersection

• Inconsistent System: No solution exists

2 Infinitely Many Solutions (Coinciding Lines)
Exercise 2
Solve the system of equations:
\(x + 2y = 4\) and \(2x + 4y = 8\)
Show that there are infinitely many solutions and explain why.
Definition:

Dependent System: A system with infinitely many solutions; the equations represent the same line

System
x + 2y = 4, 2x + 4y = 8
Elimination
0 = 0
Conclusion
Infinitely many solutions
Step 1: Align the equations

\(x + 2y = 4\)

\(2x + 4y = 8\)

Step 2: Attempt to eliminate one variable

Multiply the first equation by 2: \(2x + 4y = 8\)

Now subtract from the second equation:

\((2x + 4y) - (2x + 4y) = 8 - 8\)

\(0 = 0\)

Step 3: Analyze the result

The statement \(0 = 0\) is always true, which is an identity

This means the equations are equivalent and represent the same line

Step 4: Verify by examining the equations

Second equation: \(2x + 4y = 8\)

Divide by 2: \(x + 2y = 4\)

This is identical to the first equation

Step 5: Conclusion

Since both equations represent the same line, every point on the line is a solution

The system has infinitely many solutions

Step 6: Express the solution set

The solution set is all points \((x, y)\) such that \(x + 2y = 4\)

Or in slope-intercept form: \(y = -\frac{1}{2}x + 2\)

Infinitely many solutions
Final answer:

The system has infinitely many solutions because the equations represent the same line. Any point on the line \(x + 2y = 4\) is a solution.

Applied rules:

• Identity: True statements like 0 = 0 indicate infinitely many solutions

• Coinciding Lines: Same equation in different forms → All points are solutions

• Dependent System: Infinitely many solutions exist

3 Identifying by Coefficients
Exercise 3
Determine whether the system has no solution, one solution, or infinitely many solutions without solving:
\(3x - 4y = 7\) and \(6x - 8y = 14\)
Justify your answer by analyzing the coefficients.
Definition:

Coefficient Analysis: Examining ratios of coefficients to determine the number of solutions

System
3x - 4y = 7, 6x - 8y = 14
Ratios
3/6 = 4/8 = 7/14
Conclusion
Infinitely many solutions
Step 1: Write equations in standard form

First equation: \(3x - 4y = 7\)

Second equation: \(6x - 8y = 14\)

Step 2: Compare coefficients using ratios

Ratio of x-coefficients: \(\frac{3}{6} = \frac{1}{2}\)

Ratio of y-coefficients: \(\frac{-4}{-8} = \frac{1}{2}\)

Ratio of constants: \(\frac{7}{14} = \frac{1}{2}\)

Step 3: Analyze the ratios

Since all three ratios are equal: \(\frac{3}{6} = \frac{-4}{-8} = \frac{7}{14} = \frac{1}{2}\)

This means the second equation is exactly \(\frac{1}{2}\) times the first equation

Step 4: Apply the classification rule

When \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\), the system is dependent

This means the equations represent the same line

Step 5: Verify by simplification

Multiply the first equation by 2: \(6x - 8y = 14\)

This is identical to the second equation

Step 6: Conclusion

The system has infinitely many solutions

Infinitely many solutions
Final answer:

The system has infinitely many solutions because the ratios of corresponding coefficients are equal, indicating that the equations represent the same line.

Applied rules:

• Coefficient Ratios: Compare \(\frac{a_1}{a_2}\), \(\frac{b_1}{b_2}\), \(\frac{c_1}{c_2}\)

• Dependent System: All ratios equal → infinitely many solutions

• Inconsistent System: First two ratios equal, third different → no solution

Special Cases of Systems Rules and Methods
\(\begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}\)
System of Linear Equations
One Solution
Different slopes
Lines intersect at one point
No Solution
Same slope, diff. y-int
Parallel lines
Infinitely Many
Same equation
Coinciding lines
Key definitions:

Consistent System: A system with at least one solution (one or infinitely many)

Inconsistent System: A system with no solutions

Independent System: A consistent system with exactly one solution

Dependent System: A consistent system with infinitely many solutions

Complete methodology:
  1. Attempt Solution: Try to solve using elimination or substitution
  2. Analyze Result: Check if you get a contradiction, identity, or specific solution
  3. Examine Coefficients: Compare ratios of corresponding coefficients
  4. Verify Graphically: Consider the geometric relationship of the lines
  5. Classify System: Determine if consistent/independent, consistent/dependent, or inconsistent
Tip 1: If elimination results in 0 = non-zero number, there's no solution.
Tip 2: If elimination results in 0 = 0, there are infinitely many solutions.
Tip 3: For coefficient analysis, compare ratios: \(\frac{a_1}{a_2}\), \(\frac{b_1}{b_2}\), \(\frac{c_1}{c_2}\).
Tip 4: Always verify your classification by checking the relationship between equations.
Common errors: Confusing no solution with infinitely many solutions, misinterpreting coefficient ratios, not checking the final result.
Exam preparation: Practice identifying each case quickly, memorize the coefficient ratio rules, work with word problems.
Formulas to know by heart:

• One Solution: \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\) (different slopes)

• No Solution: \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\) (parallel lines)

• Infinitely Many: \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\) (same line)

• Contradiction: False statement indicates no solution

• Identity: Always true statement indicates infinitely many solutions

Solution: Exercises 4 to 5
4 Word Problem - No Solution
Exercise 4
A store sells pens for $2 each and notebooks for $3 each. On Monday, a customer bought 5 items for $15. On Tuesday, another customer bought 5 items for $20. Write a system of equations and determine if there's a consistent price for pens and notebooks.
Definition:

Real-World Contradiction: When a system derived from real-world data has no solution, it indicates inconsistent information

Variables
x=pen price, y=notebook price
System
x + y = 3, x + y = 4
Conclusion
No solution
Step 1: Define variables

Let \(x\) = price of one pen, \(y\) = price of one notebook

Step 2: Set up equations from Monday's purchase

Monday: 5 items for $15

This could be represented by various combinations, but let's say \(a\) pens and \(b\) notebooks

For example: 2 pens and 3 notebooks → \(2x + 3y = 15\) and \(2 + 3 = 5\)

Or: 3 pens and 2 notebooks → \(3x + 2y = 15\) and \(3 + 2 = 5\)

Step 3: Set up equations from Tuesday's purchase

Tuesday: 5 items for $20

Same logic: 2 pens and 3 notebooks → \(2x + 3y = 20\) and \(2 + 3 = 5\)

Step 4: Compare the equations

If both customers bought the same combination (2 pens, 3 notebooks):

Monday: \(2x + 3y = 15\)

Tuesday: \(2x + 3y = 20\)

Step 5: Attempt to solve

Subtract first from second: \((2x + 3y) - (2x + 3y) = 20 - 15\)

\(0 = 5\)

This is a contradiction

Step 6: Conclusion

The system has no solution, which means the prices must have changed between Monday and Tuesday, or the item counts are incorrect

No solution (prices changed)
Final answer:

The system has no solution, indicating that the prices of pens and notebooks changed between Monday and Tuesday.

Applied rules:

• Real-World Modeling: Translate word problems into mathematical equations

• Contradiction Interpretation: No solution indicates inconsistent real-world data

• Problem Analysis: Use mathematical results to draw real-world conclusions

5 Word Problem - Infinitely Many Solutions
Exercise 5
A recipe calls for flour and sugar in a ratio of 2:1. Another recipe calls for twice as much flour and twice as much sugar as the first recipe. Write a system of equations and determine the possible amounts of flour and sugar.
Definition:

Proportional Relationships: When one equation is a multiple of another, infinitely many solutions exist

Variables
x=flour, y=sugar
System
x = 2y, 2x = 4y
Conclusion
Infinitely many solutions
Step 1: Define variables

Let \(x\) = amount of flour, \(y\) = amount of sugar

Step 2: Set up equation for first recipe

Ratio of flour to sugar is 2:1 → \(\frac{x}{y} = \frac{2}{1}\)

So: \(x = 2y\)

Step 3: Set up equation for second recipe

Second recipe uses twice as much of each → \(2x\) flour and \(2y\) sugar

But this is still in the ratio 2:1 → \(\frac{2x}{2y} = \frac{2}{1}\)

This simplifies to: \(\frac{x}{y} = \frac{2}{1}\), which is the same as \(x = 2y\)

Step 4: Write the system

\(\begin{cases} x = 2y \\ x = 2y \end{cases}\)

Or equivalently: \(\begin{cases} x - 2y = 0 \\ x - 2y = 0 \end{cases}\)

Step 5: Attempt to solve

Subtracting the equations: \((x - 2y) - (x - 2y) = 0 - 0\)

\(0 = 0\)

This is an identity, indicating infinitely many solutions

Step 6: Express the solution set

Any values where \(x = 2y\) satisfy both equations

For example: (2,1), (4,2), (6,3), (10,5), etc.

Infinitely many solutions: x = 2y
Final answer:

The system has infinitely many solutions where the amount of flour is twice the amount of sugar (x = 2y).

Applied rules:

• Proportional Reasoning: When one equation is a scalar multiple of another

• Identity Interpretation: 0 = 0 indicates dependent equations

• Parameterized Solutions: Express solutions in terms of one variable

Special Cases of Systems Fundamentals
\(\begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}\)
System of Linear Equations
Key definitions:

Consistent System: A system that has at least one solution (either one solution or infinitely many solutions)

Inconsistent System: A system that has no solutions

Independent System: A consistent system with exactly one solution

Dependent System: A consistent system with infinitely many solutions

Complete methodology:
  1. Attempt to Solve: Use elimination or substitution method
  2. Analyze Result: Check if you get a specific solution, contradiction, or identity
  3. Examine Coefficients: Compare ratios of corresponding coefficients
  4. Graphical Interpretation: Consider the geometric relationship
  5. Verify Conclusion: Double-check your classification
Tip 1: When using elimination, watch for 0 = non-zero (no solution) or 0 = 0 (infinitely many).
Tip 2: In slope-intercept form, same slope with different y-intercept = no solution.
Tip 3: In slope-intercept form, same slope with same y-intercept = infinitely many solutions.
Tip 4: Always verify your conclusion by checking the original equations.
Applications: Economics (supply/demand analysis), physics (motion problems), engineering (circuit analysis), business (break-even analysis).
Properties: No solution = parallel lines; infinitely many = same line; one solution = intersecting lines.
Essential formulas:

• One Solution: \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\) (different slopes)

• No Solution: \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\) (parallel lines)

• Infinitely Many: \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\) (same line)

• Algebraic Test: Contradiction = no solution, Identity = infinitely many

Types of Systems Visualization
Exercise 6: System Types Comparison
Compare these systems:
One solution: y = x + 1 and y = -x + 3
No solution: y = 2x + 1 and y = 2x + 3
Infinitely many: y = x + 1 and 2y = 2x + 2

Analysis: The chart shows how different systems of equations have different graphical representations.

  • One solution: Lines intersect at exactly one point
  • No solution: Lines are parallel and never meet
  • Infinitely many: Lines are identical and overlap completely

Questions & Answers

Question: How can I tell if a system has no solution or infinitely many solutions just by looking at the equations?

Answer: You can analyze the coefficients in standard form \(a_1x + b_1y = c_1\) and \(a_2x + b_2y = c_2\):

No Solution Case: The ratios of coefficients are equal for x and y but not for the constants:

  • \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\)
  • Example: \(2x + 3y = 6\) and \(4x + 6y = 10\) (ratios: 2/4 = 3/6 ≠ 6/10)

Infinitely Many Solutions Case: All ratios are equal:

  • \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\)
  • Example: \(2x + 3y = 6\) and \(4x + 6y = 12\) (ratios: 2/4 = 3/6 = 6/12 = 1/2)

This happens when the equations represent parallel lines (no solution) or the same line (infinitely many solutions).

Question: What does it mean when I get 0 = 0 while solving a system?

Answer: Getting \(0 = 0\) means the system has infinitely many solutions:

  • The statement \(0 = 0\) is always true, which is called an identity
  • This happens when both equations represent the same line
  • Every point on the line is a solution to the system
  • The system is called "dependent" or "consistent and dependent"

For example, if you solve the system:

  • \(x + y = 5\)
  • \(2x + 2y = 10\)

Multiplying the first equation by 2 gives the second equation, so when you subtract them, you get \(0 = 0\). This means every point on the line \(x + y = 5\) is a solution.

Question: Why do inconsistent systems occur in real-world problems?

Answer: Inconsistent systems in real-world problems indicate:

  • Data Errors: Incorrect measurements or data entry
  • Changing Conditions: Parameters that change between observations
  • Over-Constrained Problems: Too many conflicting requirements
  • Impossible Scenarios: Situations that cannot exist simultaneously

For example, if a business model shows that producing 100 units costs $500 in one scenario and $600 in another identical scenario, this inconsistency indicates that some variable (like material costs or labor rates) has changed.

The mathematical inconsistency alerts us that our model doesn't match reality, prompting investigation into what assumptions or data might be wrong.

Question: How do I express the solution when there are infinitely many solutions?

Answer: There are several ways to express infinitely many solutions:

1. Parametric Form: Express one variable in terms of the other

  • If the line is \(y = 2x + 3\), the solution is \((x, 2x + 3)\) for any real number \(x\)

2. Set Builder Notation: \(\{(x, y) | y = 2x + 3\}\)

3. Verbal Description: "All points on the line \(y = 2x + 3\)"

4. Specific Examples: Give a few specific solutions like \((0, 3)\), \((1, 5)\), \((-1, 1)\)

The most common approach is to solve for one variable in terms of the other and state that the solution is all points satisfying that relationship.

For example, if both equations reduce to \(x + 2y = 4\), the solution is all points \((x, y)\) such that \(x + 2y = 4\), or equivalently \(y = -\frac{1}{2}x + 2\).