Systems of Linear Equations

A system of linear equations is a set of two or more linear equations considered at the same time.

For example:

y = 2x + 1

y = −x + 7

Each equation represents a straight line. When both equations are graphed on the same coordinate plane, the lines may intersect.

The point where they intersect is especially important because it satisfies both equations at the same time.

For this system, the lines intersect at:

(2, 5)

Check the first equation:

5 = 2(2) + 1

5 = 5 ✓

Check the second equation:

5 = −2 + 7

5 = 5 ✓

Therefore:

(2, 5) is the solution to the system.

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What Is a System?

A system is a collection of equations that describe related conditions.

A simple system might be written as:

y = x + 2

y = −2x + 8

We are looking for values of x and y that make both equations true simultaneously.

This is different from solving one equation.

With one equation such as:

y = x + 2

there are infinitely many ordered pairs that satisfy it:

(0, 2)

(1, 3)

(2, 4)

(3, 5)

and many more.

A system asks:

Which ordered pair belongs to both lines?

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The Graphical Meaning of a System

Every linear equation represents a line.

When two equations form a system, graph both lines on the same coordinate plane.

The solution is the point that belongs to both graphs.

In other words:

solution to system = intersection of the graphs

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Why the Intersection Is the Solution

Suppose:

y = x + 3

and:

y = −x + 7

The lines intersect at:

(2, 5)

For the first equation:

5 = 2 + 3

True.

For the second:

5 = −2 + 7

True.

Therefore, the point:

(2, 5)

satisfies both equations.

That is why an intersection represents the solution to a system.

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Ordered Pairs as Solutions

A solution to a two-variable system is normally written as an ordered pair:

(x, y)

For example:

(3, 7)

means:

x = 3

y = 7

For this point to solve a system, it must satisfy every equation in the system.

Satisfying only one equation is not enough.

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Testing a Possible Solution

Consider:

y = 2x + 1

y = x + 4

Is:

(3, 7)

a solution?

First equation:

7 = 2(3) + 1

7 = 7 ✓

Second equation:

7 = 3 + 4

7 = 7 ✓

The point satisfies both equations.

Therefore:

(3, 7) is a solution to the system.

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A Point That Satisfies Only One Equation

Consider the same system:

y = 2x + 1

y = x + 4

Test:

(2, 5)

First equation:

5 = 2(2) + 1

5 = 5 ✓

Second equation:

5 = 2 + 4

5 = 6 ✗

The point lies on the first line but not the second.

Therefore:

(2, 5) is not a solution to the system.

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Solving a System by Graphing

To solve a system graphically:

1. Graph the first equation.
2. Graph the second equation on the same coordinate plane.
3. Locate the intersection.
4. Read its coordinates.
5. Write the solution as (x, y).
6. Substitute the coordinates into both equations to check.

This method is called solving by graphing.

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Graphing from Slope-Intercept Form

Equations written as:

y = mx + b

are particularly convenient for graphing.

Remember:

m = slope

b = y-intercept

Consider:

y = x + 1

y = −x + 5

For the first line:

m = 1

b = 1

For the second:

m = −1

b = 5

Graph both lines and locate their intersection.

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Worked Example

Solve:

y = x + 1

y = −x + 5

Create some points.

For:

y = x + 1

For:

y = −x + 5

Both tables contain:

(2, 3)

Therefore, the graphs intersect at:

(2, 3)

The solution is:

(2, 3)

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Checking the Solution

Substitute:

x = 2

y = 3

into:

y = x + 1

3 = 2 + 1

3 = 3 ✓

Now check:

y = −x + 5

3 = −2 + 5

3 = 3 ✓

The solution is confirmed.

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Graphing with Slope and Intercept

Consider:

y = 2x − 1

y = −x + 5

For the first equation:

m = 2

b = −1

Plot:

(0, −1)

Then use slope:

2 = 2/1

Move 1 right and 2 up.

For the second:

m = −1

b = 5

Plot:

(0, 5)

Then move 1 right and 1 down.

The lines intersect at:

(2, 3)

Therefore:

solution = (2, 3)

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Reading an Intersection from a Graph

Suppose two lines intersect at:

(4, 6)

Then the solution is:

x = 4

y = 6

This means that when x equals 4, both relationships produce y = 6.

The meaning of these values depends on the context.

For example:

• 4 hours and $6
• 4 items and $6
• 4 kilometres and 6 litres

The axes and units determine the interpretation.

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Three Possible Types of Systems

When two linear equations are graphed, there are three basic possibilities:

• the lines intersect once
• the lines never intersect
• the lines are actually the same line

These produce different numbers of solutions.

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One Solution

Consider:

y = x + 2

y = −2x + 8

The slopes are different:

1 and −2

Therefore, the lines eventually intersect.

The system has:

one solution

This type of system is often called a consistent independent system.

For introductory work, the important idea is:

different slopes → one intersection → one solution

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No Solution

Consider:

y = 2x + 1

y = 2x + 5

Both lines have slope:

2

but their y-intercepts are different:

1 and 5

Therefore, the lines are parallel.

Parallel lines never intersect.

The system has:

no solution

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Why Parallel Lines Have No Solution

Consider:

y = 3x + 2

and:

y = 3x − 4

The lines increase at exactly the same rate.

Their vertical separation remains constant.

No ordered pair lies on both lines.

Therefore, there is no value of x and y that makes both equations true simultaneously.

The solution set is:

∅

where ∅ represents the empty set.

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Infinitely Many Solutions

Consider:

y = 2x + 3

and:

2y = 4x + 6

Simplify the second equation:

2y = 4x + 6

Divide by 2:

y = 2x + 3

The two equations represent exactly the same line.

Every point on one line is also on the other.

Therefore, the system has:

infinitely many solutions

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Comparing the Three Cases

This allows us to predict the number of solutions before graphing in many cases.

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Intersection Does Not Mean the Lines Are Equal Everywhere

Suppose two lines intersect at:

(3, 8)

At that particular point, both equations produce the same output.

Away from the intersection, the outputs are generally different.

Therefore, the intersection means:

the relationships are equal at one particular input

It does not mean the equations describe the same relationship everywhere.

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Systems from Tables

Systems can also be recognized using tables.

Consider:

Relationship A

Relationship B

Both relationships contain:

(3, 8)

Therefore, their graphs intersect at:

(3, 8)

This is the solution to the system.

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Systems in Real-World Situations

Systems are useful whenever we want to compare two relationships.

Common examples include:

• comparing phone plans
• comparing taxi companies
• comparing wages
• comparing rental costs
• comparing savings plans
• comparing production methods
• determining when two moving objects reach the same position
• finding when two quantities become equal

The intersection represents the point at which the two situations have the same output for the same input.

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Comparing Two Taxi Companies

Suppose two taxi companies charge:

Company A: C = 3d + 5

Company B: C = 2d + 10

where:

C = cost in dollars

d = distance in kilometres

We want to know:

At what distance do the two companies charge the same amount?

This question creates a system:

C = 3d + 5

C = 2d + 10

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Interpreting the Taxi Graph

Company A has:

slope = 3 dollars/km

intercept = $5

Company B has:

slope = 2 dollars/km

intercept = $10

Company A starts cheaper but increases more quickly.

Company B starts more expensive but increases more slowly.

Their graphs eventually intersect.

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Finding the Taxi Intersection

A graph of the two equations shows that they intersect at:

(5, 20)

In context:

d = 5 km

C = $20

Therefore:

At a distance of 5 km, both companies charge $20.

The ordered pair must be interpreted using the variables rather than simply reported as numbers.

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Comparing Before and After the Intersection

At distances below 5 km:

Company A is cheaper.

At exactly 5 km:

both cost $20.

At distances greater than 5 km:

Company B is cheaper.

The intersection can therefore act as a break-even point.

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Break-Even Points

A break-even point occurs when two quantities are equal.

For example, a business might compare:

revenue

and:

cost

The point where the graphs intersect represents the amount of production or sales at which:

revenue = cost

Before this point, the business may have a loss.

After this point, it may have a profit, depending on the model.

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Real-World Example: Gym Memberships

Gym A charges:

A = 8v + 20

Gym B charges:

B = 5v + 35

where:

v = number of visits

cost is measured in dollars.

The graphs intersect at:

v = 5

Check:

A = 8(5) + 20

A = 60

B = 5(5) + 35

B = 60

Therefore, the intersection is:

(5, 60)

Interpretation:

At 5 visits, both gyms cost $60.

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Deciding Which Option Is Better

The intersection alone does not always answer the entire question.

For the gym example:

Gym A starts at $20 but costs $8 per visit.

Gym B starts at $35 but costs $5 per visit.

For fewer than 5 visits:

Gym A is cheaper.

At 5 visits:

the costs are equal.

For more than 5 visits:

Gym B is cheaper.

Systems therefore help us compare alternatives.

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Real-World Example: Savings

Suppose:

Mia starts with $100 and saves $25 each week.

M = 25w + 100

Noah starts with $220 and saves $10 each week.

N = 10w + 220

We can ask:

When will they have the same amount of savings?

The system is:

S = 25w + 100

S = 10w + 220

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Interpreting the Savings Intersection

The graphs intersect at:

(8, 300)

This means:

After 8 weeks, both people have $300.

Before 8 weeks, Noah has more.

After 8 weeks, Mia has more because her savings increase at the greater rate.

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Real-World Example: Moving Objects

Suppose two cyclists' positions are represented by:

A: d = 12t + 5

B: d = 8t + 25

where:

d = distance from a reference point in kilometres

t = time in hours

The intersection tells us when the cyclists are at the same position at the same time.

This is important: having the same distance at different times would not solve the system.

Both coordinates must match simultaneously.

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Creating a System from a Situation

Suppose:

Company A charges a $30 starting fee plus $4 per item.

Company B charges a $50 starting fee plus $2 per item.

Define:

n = number of items

C = total cost

Company A:

C = 4n + 30

Company B:

C = 2n + 50

Together:

C = 4n + 30

C = 2n + 50

This is a system of linear equations.

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Estimating Solutions from Graphs

Not every graphical solution falls exactly on a grid intersection.

Suppose two lines appear to intersect near:

(3.4, 7.2)

The graphical solution might be estimated as:

x ≈ 3.4

y ≈ 7.2

Graphing can therefore produce approximate solutions.

The accuracy depends on:

• graph scale
• plotting accuracy
• line thickness
• how precisely the intersection can be read

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Exact and Approximate Solutions

A graph is excellent for understanding what a solution means.

However, graphical solutions are sometimes approximate.

If the intersection is:

(2, 5)

it may be easy to read exactly.

If it is:

(2.67, 4.83)

the graph may only provide an estimate.

Later algebraic methods such as substitution and elimination can often determine solutions more precisely.

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Choosing an Appropriate Scale

A graph should use a scale that makes the intersection visible.

Suppose two businesses have costs between:

$1,000 and $5,000.

A y-axis extending only from:

0 to 100

would be useless.

Likewise, a graph extending to:

1,000,000

might compress the lines so much that the intersection becomes difficult to read.

Choose a scale appropriate to the data.

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Realistic Domains

Real-world systems often have restricted domains.

Suppose x represents the number of tickets purchased.

Then:

x ≥ 0

and x should normally be a whole number.

An intersection at:

(4.5, 60)

might be mathematically correct but unrealistic if x represents people, buses, or tickets.

Always interpret the solution in context.

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An Intersection Outside the Realistic Domain

Suppose two pricing models intersect at:

x = −5

Mathematically, the lines intersect.

But if x represents the number of products purchased:

−5 products

has no practical meaning.

Therefore, the system may have a mathematical solution but no meaningful break-even point within the realistic domain.

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Checking a Graphical Solution

Suppose a graph suggests:

(4, 11)

as the solution to:

y = 2x + 3

y = −x + 15

Check the first:

11 = 2(4) + 3

11 = 11 ✓

Check the second:

11 = −4 + 15

11 = 11 ✓

Therefore, the graphical solution is confirmed.

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Why Checking Matters

Graphs can contain errors caused by:

• plotting the wrong intercept
• using the wrong slope
• misreading the scale
• drawing a line inaccurately
• estimating the intersection incorrectly

Substitution provides a quick way to verify whether the proposed ordered pair actually solves both equations.

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Systems and Different Representations

A system does not always begin as two equations.

One relationship might be shown as:

• a graph
• a table
• an equation
• a verbal description

For example:

Relationship A: y = 3x + 2

Relationship B might be represented by:

From the table, Relationship B follows:

y = −x + 8

Now the two relationships can be graphed as a system.

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Finding the Intersection from Mixed Representations

Using:

y = 3x + 2

and:

y = −x + 8

the graphs intersect between x = 1 and x = 2.

Graphing carefully gives approximately:

(1.5, 6.5)

Check:

First equation:

3(1.5) + 2 = 6.5

Second equation:

−1.5 + 8 = 6.5

Therefore:

(1.5, 6.5)

is the solution.

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Systems as Simultaneous Conditions

Another useful way to understand a system is as a set of conditions that must all be true at the same time.

For:

y = 2x + 4

y = −x + 10

a solution must satisfy:

the first condition AND the second condition.

This is why the intersection is important.

It is the point where both conditions are satisfied simultaneously.

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Systems vs Inequalities

A system of equations asks for points that lie on both lines.

A system of inequalities asks for points that lie in all required solution regions.

For equations, the solution may be:

one intersection point

For inequalities, the solution may be:

an entire feasible region

This distinction becomes important in later applications.

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Predicting the Number of Solutions

Before graphing, compare the slopes and intercepts.

For:

y = 2x + 4

y = −3x + 9

different slopes → one solution

For:

y = 5x + 2

y = 5x − 6

same slope, different intercepts → no solution

For:

y = 4x + 7

2y = 8x + 14

simplify the second equation:

y = 4x + 7

same equation → infinitely many solutions

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A Complete Graphing Example

Solve:

y = 2x

y = −x + 6

For the first line:

For the second:

Both contain:

(2, 4)

Therefore:

solution = (2, 4)

Check:

4 = 2(2) ✓

4 = −2 + 6 ✓

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A Complete Real-World Example

Two bicycle rental companies have the following prices.

Company A:

$15 starting fee plus $6 per hour.

Company B:

$27 starting fee plus $4 per hour.

Define:

h = hours rented

C = total cost

The system is:

C = 6h + 15

C = 4h + 27

The graphs intersect at:

(6, 51)

Therefore:

After 6 hours, both companies cost $51.

Before 6 hours, Company A is cheaper.

After 6 hours, Company B is cheaper.

The intersection provides both a mathematical solution and useful information for making a decision.

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A Reliable Strategy for Solving by Graphing

When solving a system graphically:

1. Write both equations clearly.
2. Rearrange them into y = mx + b if necessary.
3. Identify the slope and y-intercept of each line.
4. Choose an appropriate coordinate-plane scale.
5. Graph the first line accurately.
6. Graph the second line on the same axes.
7. Locate the intersection.
8. Read the coordinates carefully.
9. Write the solution as (x, y).
10. Substitute the point into both equations.
11. Determine whether the solution is exact or approximate.
12. If there is no intersection, state no solution.
13. If the graphs are the same line, state infinitely many solutions.
14. If the problem has context, interpret both coordinates with appropriate units.

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A Reliable Strategy for Real-World Systems

When a real situation contains two relationships:

1. Identify the two situations being compared.
2. Define the variables.
3. Include appropriate units.
4. Write an equation for each relationship.
5. Graph both equations using the same axes.
6. Find their intersection.
7. Interpret the x-coordinate.
8. Interpret the y-coordinate.
9. Determine what happens before the intersection.
10. Determine what happens after the intersection.
11. Check whether the solution is realistic within the context.
12. Verify the solution in both equations.

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Common Mistakes

Thinking every point on either line solves the system

A solution must lie on both lines.

Giving only the x-coordinate

A system solution is normally an ordered pair:

(x, y)

Switching coordinate order

The solution:

(3, 7)

means x = 3 and y = 7, not the reverse.

Reading the intersection inaccurately

Check the graph scale carefully.

Graphing the lines on separate coordinate planes

To see the intersection, both equations must be graphed on the same axes.

Assuming every system has one solution

Systems may have:

• one solution
• no solution
• infinitely many solutions

Thinking parallel lines have infinitely many solutions

Parallel distinct lines never intersect, so they have no solution.

Thinking identical lines have one solution

Identical lines share every point, so they have infinitely many solutions.

Ignoring context

An intersection at x = −4 may be mathematically valid but meaningless if x represents the number of tickets purchased.

Interpreting only one coordinate

In real situations, both coordinates usually carry important information.

Assuming graphical answers are always exact

Some intersections must be estimated from the graph.

Failing to check the solution

Substitution into both equations confirms whether the point is correct.

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Key Terms

System of linear equations — two or more linear equations considered simultaneously.

Solution of a system — value or ordered pair that satisfies every equation in the system.

Intersection — point shared by two graphs.

Ordered pair — pair of coordinates written as (x, y).

Graphical solution — solution found by locating the intersection of graphs.

One-solution system — system whose graphs intersect at exactly one point.

No-solution system — system whose graphs never intersect.

Infinitely-many-solutions system — system in which the equations represent the same graph.

Parallel lines — lines with the same slope and different intercepts that never intersect.

Coincident lines — different-looking equations that represent the same line.

Break-even point — point at which two quantities, such as costs and revenues, are equal.

Slope — rate of change of a linear relationship.

y-intercept — value of y when x = 0.

Simultaneous — occurring or being true at the same time.

Realistic domain — input values that are meaningful within a particular real-world situation.

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Key Takeaways

• A system of linear equations contains two or more linear equations considered together.
• A solution must satisfy every equation in the system simultaneously.
• Graphically, a solution is an intersection shared by the lines.
• A solution to a two-variable system is usually written as an ordered pair.
• A point lying on only one line does not solve the system.
• Systems can be solved graphically by plotting all equations on the same coordinate plane.
• Substitution can be used to verify a graphical solution.
• Two lines with different slopes have exactly one intersection.
• Parallel distinct lines have no solution.
• Identical lines have infinitely many solutions.
• Comparing slopes and intercepts can often predict the number of solutions before graphing.
• Graphical solutions may be exact or approximate depending on the intersection and graph scale.
• Systems are useful for comparing two real-world relationships.
• An intersection can represent equal costs, equal savings, the same position, or a break-even point.
• Both coordinates of a real-world intersection should be interpreted.
• What happens before and after an intersection may be just as important as the intersection itself.
• Realistic domains can restrict whether a mathematical intersection is meaningful.
• Systems may be represented using equations, tables, graphs, or verbal descriptions.
• Graphing provides a strong visual understanding of systems, while later algebraic methods can provide more precise solutions.
• The central idea is simple: the solution is where all of the conditions are true at the same time.

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Check Your Understanding

1. Explain in your own words what a system of linear equations is.
2. Explain why the intersection of two lines represents the solution to a system.
3. Determine whether (2, 7) is a solution to the system:

y = 3x + 1

y = x + 5

Show how you know.

4. Graph the system and determine its solution:

y = x + 2

y = −x + 6

5. Solve graphically:

y = 2x − 1

y = −x + 8

Then verify the solution using substitution.

6. Without graphing, predict whether the system has one solution, no solution, or infinitely many solutions:

y = 4x + 2

y = 4x − 7

Explain your reasoning.

7. Determine the number of solutions:

y = −2x + 5

2y = −4x + 10

Explain how you know.

8. One relationship contains the points (0, 3) and (2, 7). Another contains (0, 9) and (2, 7). What is the graphical solution to the system? Explain what the shared point means.
9. Two taxi companies use the models:

A: C = 4d + 6

B: C = 2d + 14

Graph the equations, determine their intersection, and interpret both coordinates in context.

10. Two savings plans are represented by:

A = 30w + 80

B = 15w + 200

Determine when the savings amounts are equal. Explain which plan has more money before and after this point.

11. Explain why two distinct parallel lines cannot have a solution.
12. A system's graph appears to intersect near (3.2, 8.7). Explain why this may be an approximate rather than exact solution and describe how you could check whether the estimate is reasonable.
13. A system describing ticket costs has an intersection at (−5, 20). Explain why this might be a valid mathematical intersection but not a meaningful solution in the real-world situation.
14. Two rental companies charge:

Company A: C = 7h + 18

Company B: C = 4h + 36

Determine the intersection. Interpret it in context and explain which company is cheaper before and after the intersection.

15. A school is choosing between two transportation companies. Company A charges a fixed fee of $250 plus $12 per student. Company B charges a fixed fee of $400 plus $7 per student. Define the variables, create a system of linear equations, determine the intersection, interpret both coordinates, and explain how the school could use the result to decide which company is less expensive for different group sizes.