5. Types of Solutions

Types of Solutions

A system of two linear equations does not always have exactly one solution. Depending on how the two equations are related, a system can have:

• One solution
• No solution
• Infinitely many solutions

When the equations are graphed, these three possibilities can be recognized by looking at the relationship between the two lines.

Learning Targets

• I can identify systems with one solution.
• I can recognize systems with no solution.
• I can recognize systems with infinitely many solutions.
• I can relate graphical representations to solution types.
• I can explain why different systems have different numbers of solutions.

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What Is a Solution to a System?

A solution to a system of equations is a set of values that satisfies both equations at the same time.

For two linear equations involving x and y, a solution is usually written as an ordered pair:

(x, y)

Graphically, a solution is a point that lies on both lines.

This means the number of solutions depends on how the two lines are positioned relative to each other.

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The Three Possibilities

For two linear equations, there are three possible graphical relationships.

The lines can:

• Intersect once
• Never intersect
• Lie exactly on top of each other

These correspond to:

Intersecting lines → One solution

Parallel lines → No solution

Same line → Infinitely many solutions

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One Solution

A system has one solution when the two lines intersect at exactly one point.

For example:

y = x + 1

y = −x + 5

The first line has slope:

1

The second line has slope:

−1

Because their slopes are different, the lines travel in different directions and intersect once.

Their intersection is:

(2, 3)

Therefore:

Solution = (2, 3)

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Why Intersecting Lines Have One Solution

Every point on the first line satisfies the first equation.

Every point on the second line satisfies the second equation.

Only at their intersection does the same ordered pair satisfy both equations.

Therefore, the system has exactly one solution.

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Different Slopes Mean One Solution

Consider equations written in slope-intercept form:

y = m₁x + b₁

y = m₂x + b₂

If:

m₁ ≠ m₂

the lines have different slopes.

Two non-vertical lines with different slopes will eventually intersect at exactly one point.

Therefore:

Different slopes → One solution

For example:

y = 2x + 3

y = 5x − 6

The slopes are:

2

and:

5

Therefore, the system has one solution.

We can know the type of solution without even finding the actual intersection.

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Worked Example — One Solution

Determine the number of solutions:

y = 3x + 2

y = −2x + 7

Compare the slopes:

m₁ = 3

m₂ = −2

The slopes are different.

Therefore, the lines intersect once.

The system has one solution.

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No Solution

A system has no solution when there is no ordered pair that satisfies both equations.

Graphically, this occurs when the lines are parallel.

Consider:

y = 2x + 1

y = 2x + 5

Both lines have slope:

2

But their y-intercepts are different:

1

and:

5

The lines have the same steepness and direction but cross the y-axis at different locations.

They never meet.

Therefore:

No solution

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Why Parallel Lines Have No Solution

For a point to be a solution, it must lie on both lines.

Parallel lines never intersect.

Therefore, there is no point that belongs to both graphs.

This means there is no ordered pair that satisfies both equations simultaneously.

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Same Slope, Different Intercepts

Suppose:

y = mx + b₁

and:

y = mx + b₂

If:

b₁ ≠ b₂

then the lines have the same slope but different y-intercepts.

They are parallel.

Therefore:

Same slope + different intercepts → No solution

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Worked Example — No Solution

Determine the number of solutions:

y = −3x + 4

y = −3x − 2

Compare the slopes:

−3 and −3

They are the same.

Compare the y-intercepts:

4 and −2

They are different.

Therefore, the lines are parallel.

The system has no solution.

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Infinitely Many Solutions

A system has infinitely many solutions when both equations represent exactly the same line.

Consider:

y = 2x + 4

and:

2y = 4x + 8

At first, the equations look different.

Simplify the second equation by dividing every term by 2:

y = 2x + 4

Now we can see that the equations are identical.

Every point on the first line also lies on the second line.

Therefore:

Infinitely many solutions

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Why the Same Line Has Infinitely Many Solutions

A line contains infinitely many points.

If two equations describe the same line, every point on that line satisfies both equations.

For example:

(0, 4)

(1, 6)

(2, 8)

and infinitely many other points all satisfy:

y = 2x + 4

If both equations represent this same line, all of these points are solutions to the system.

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Same Slope and Same Intercept

If two equations simplify to:

y = mx + b

with the same m and the same b, they describe the same line.

Therefore:

Same slope + same intercept → Infinitely many solutions

However, the equations may need to be simplified before this becomes obvious.

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Worked Example — Infinitely Many Solutions

Determine the number of solutions:

y = 3x + 6

2y = 6x + 12

Simplify the second equation:

2y = 6x + 12

Divide everything by 2:

y = 3x + 6

Both equations are now identical.

Therefore:

The system has infinitely many solutions.

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Equations Can Look Different but Represent the Same Line

Consider:

x + y = 5

and:

2x + 2y = 10

Divide the second equation by 2:

x + y = 5

The equations are equivalent.

Therefore, they represent the same line.

The system has:

Infinitely many solutions

This is why it is important to simplify equations before comparing them.

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Comparing Systems Using Slope-Intercept Form

Writing equations in:

y = mx + b

form makes solution types easier to recognize.

For example:

2x + y = 7

Rearrange:

y = −2x + 7

Now compare with:

y = −2x + 3

Both slopes are:

−2

but the intercepts are different.

Therefore:

No solution

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Worked Example — Rearranging Before Comparing

Determine the type of solution:

3x + y = 8

y = −3x + 2

Rearrange the first equation:

y = −3x + 8

Now compare:

y = −3x + 8

y = −3x + 2

The slopes are equal.

The intercepts are different.

Therefore:

No solution

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Worked Example — Hidden Equivalent Equations

Determine the type of solution:

2x + 4y = 12

x + 2y = 6

Multiply the second equation by 2:

2x + 4y = 12

This is exactly the first equation.

Therefore, both equations represent the same line.

The system has infinitely many solutions.

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A Quick Comparison

For equations in slope-intercept form:

y = m₁x + b₁

and:

y = m₂x + b₂

remember:

m₁ ≠ m₂ → One solution

m₁ = m₂ but b₁ ≠ b₂ → No solution

m₁ = m₂ and b₁ = b₂ → Infinitely many solutions

This provides a fast algebraic way to classify many systems without graphing them.

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Recognizing Solution Types from a Graph

If you are given a graph rather than equations, look at the relationship between the lines.

If the lines cross:

One solution

If the lines remain parallel:

No solution

If only one line appears because both graphs overlap completely:

Infinitely many solutions

Graphing gives us a visual way to classify systems.

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Recognizing Solution Types from Equations

When given equations:

• Simplify the equations if necessary.
• Rewrite them in slope-intercept form when useful.
• Compare their slopes.
• Compare their intercepts.

Ask:

Are the slopes different?

If yes:

One solution

If the slopes are the same, ask:

Are the intercepts different?

If yes:

No solution

If both slope and intercept are the same:

Infinitely many solutions

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Recognizing Solution Types Without Graphing

Consider:

y = 4x − 7

y = 4x + 3

You do not need to graph these equations.

The slopes are the same:

4

The intercepts are different:

−7 and 3

Therefore:

No solution

Recognizing these patterns can save considerable time.

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What Happens When We Solve Algebraically?

The three types of solutions also appear when we use algebra.

Sometimes solving produces a value such as:

x = 3

This indicates that there is a specific solution.

But sometimes the variables disappear.

The statement that remains can tell us whether there are no solutions or infinitely many solutions.

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When Algebra Produces a False Statement

Consider:

y = 2x + 1

y = 2x + 5

Since both equal y:

2x + 1 = 2x + 5

Subtract 2x from both sides:

1 = 5

This is impossible.

There are no values of x that make this true.

Therefore:

No solution

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When Algebra Produces a True Statement

Consider:

y = 2x + 4

2y = 4x + 8

Simplify the second equation:

y = 2x + 4

If we compare the equations algebraically, everything cancels and we may obtain:

0 = 0

This statement is always true.

The equations represent the same relationship.

Therefore:

Infinitely many solutions

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Contradictions and Identities

When solving systems, two special algebraic results can appear.

A contradiction is a statement that can never be true.

For example:

3 = 8

This indicates:

No solution

An identity is a statement that is always true.

For example:

0 = 0

This indicates:

Infinitely many solutions

These results connect algebraic solving directly to the graphical relationships between lines.

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Why Different Systems Have Different Numbers of Solutions

The number of solutions depends on whether the equations describe different or identical relationships.

If the lines have different slopes, they eventually meet:

One solution

If they have the same slope but begin at different positions, they remain separated:

No solution

If they describe exactly the same relationship, they overlap completely:

Infinitely many solutions

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Real-World Meaning of One Solution

Suppose two companies have pricing equations:

Company A: C = 3x + 10

Company B: C = 5x + 2

The slopes are different.

Their graphs intersect once.

The intersection represents the quantity at which both companies charge the same price.

Before or after that point, one company may cost more than the other.

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Real-World Meaning of No Solution

Suppose:

Company A: C = 4x + 5

Company B: C = 4x + 10

Both companies increase their price at exactly the same rate:

$4 per unit

But Company B always starts $5 higher.

Their graphs are parallel.

There is no quantity at which the two companies charge the same price.

Therefore, the system has no solution.

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Real-World Meaning of Infinitely Many Solutions

Suppose:

Company A: C = 3x + 6

and Company B's pricing rule simplifies to:

C = 3x + 6

The companies are charging exactly the same amount for every possible quantity.

Their graphs are identical.

Therefore, there are:

Infinitely many solutions

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Classifying Without Finding the Actual Solution

Sometimes a question asks only:

How many solutions does this system have?

You do not necessarily need to solve the entire system.

For example:

y = 7x + 4

y = −2x + 9

The slopes are different.

Therefore:

One solution

We do not need to calculate where the intersection occurs unless the problem specifically asks for the solution.

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Worked Example — Classify the System

Consider:

y = 5x − 2

y = 5x + 8

Slope of first line:

5

Slope of second line:

5

The intercepts are:

−2

and:

8

Same slope, different intercepts.

Therefore:

No solution

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Worked Example — Another Classification

Consider:

4x + 2y = 10

2x + y = 5

Multiply the second equation by 2:

4x + 2y = 10

This matches the first equation exactly.

Therefore:

Infinitely many solutions

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Worked Example — One Solution in Standard Form

Consider:

2x + y = 7

x − y = 2

Rewrite the first equation:

y = −2x + 7

Rewrite the second:

−y = 2 − x

so:

y = x − 2

The slopes are:

−2

and:

1

Because the slopes are different:

One solution

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Connecting Graphical and Algebraic Evidence

The graphical and algebraic approaches tell the same story.

Different slopes

Graphically: the lines intersect.

Algebraically: solving produces one ordered pair.

Result:

One solution

Same slope, different intercepts

Graphically: the lines are parallel.

Algebraically: solving produces a contradiction such as 0 = 4.

Result:

No solution

Equivalent equations

Graphically: the lines overlap.

Algebraically: solving produces an identity such as 0 = 0.

Result:

Infinitely many solutions

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Common Mistakes

Mistake: Assuming Every System Has One Solution

Systems can have one, zero, or infinitely many solutions.

Always examine the relationship between the equations.

Mistake: Looking Only at the Y-Intercept

Different intercepts do not automatically mean no solution.

For example:

y = 2x + 1

and:

y = 3x + 5

have different intercepts but also different slopes.

They will intersect.

Therefore, they have one solution.

Mistake: Looking Only at the Slope

If the slopes are equal, you must also compare the intercepts.

Same slope + different intercepts:

No solution

Same slope + same intercept:

Infinitely many solutions

Mistake: Comparing Equations Before Simplifying

For example:

2y = 4x + 6

should first be simplified to:

y = 2x + 3

before comparing it with another equation.

Mistake: Thinking Overlapping Lines Have One Solution

If the lines overlap completely, every point on the line satisfies both equations.

There are:

Infinitely many solutions

Mistake: Confusing a Contradiction with an Identity

0 = 5

is impossible and indicates:

No solution

0 = 0

is always true and indicates:

Infinitely many solutions

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Key Terms

System of equations – two or more equations considered together.

Solution – values that satisfy all equations simultaneously.

One solution – exactly one ordered pair satisfies the system.

No solution – no ordered pair satisfies the system.

Infinitely many solutions – infinitely many ordered pairs satisfy the system.

Intersecting lines – lines that meet at a point.

Parallel lines – lines with the same slope that never meet.

Coincident lines – lines that lie exactly on top of each other.

Slope – the rate of change of a line.

Y-intercept – the point where a line crosses the y-axis.

Contradiction – a mathematical statement that cannot be true.

Identity – a mathematical statement that is always true.

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Check Your Understanding

1. How many solutions does a system have when two lines intersect once?

2. How many solutions does a system have when two lines are parallel?

3. How many solutions does a system have when two equations represent the same line?

4. Classify:

y = 2x + 3

y = −x + 6

5. Classify:

y = 4x + 1

y = 4x − 7

6. Classify:

y = 3x + 5

2y = 6x + 10

7. Explain why two lines with different slopes must have one solution.

8. Explain why parallel lines have no solution.

9. Explain why identical lines have infinitely many solutions.

10. Classify:

2x + y = 8

y = −2x + 3

11. Classify:

3x + 6y = 12

x + 2y = 4

12. What does a result such as:

0 = 7

tell you about a system?

13. What does a result such as:

0 = 0

tell you about a system?

14. A student says that two equations with different y-intercepts can never have a solution. Explain why this is incorrect.

15. Describe how slope and y-intercept can be used to determine the number of solutions without graphing the system.

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Key Takeaways

• A system of two linear equations can have one solution, no solution, or infinitely many solutions.
• Intersecting lines have one solution.
• Parallel lines have no solution.
• Identical or coincident lines have infinitely many solutions.
• Lines with different slopes have one solution.
• Lines with the same slope but different intercepts have no solution.
• Lines with the same slope and the same intercept represent the same line and have infinitely many solutions.
• Equations may need to be simplified before their relationship becomes obvious.
• A contradiction such as 0 = 5 indicates no solution.
• An identity such as 0 = 0 indicates infinitely many solutions.
• Graphical and algebraic methods describe the same relationships.
• You can often determine the number of solutions without finding the actual solution.
• Understanding solution types helps explain how two linear relationships compare.

Did You Know?

The three possible solution types are really descriptions of the geometry of straight lines. Two distinct lines in a plane either intersect once or are parallel, while two equations can also describe the exact same line. This simple geometric idea explains why a system of two linear equations can have one, zero, or infinitely many solutions—but never exactly two or three solutions.