Graphing Linear Equations

A linear equation in two variables describes a relationship between two quantities whose graph forms a straight line.

For example:

y = 2x + 1

contains two variables, x and y. Every ordered pair that satisfies the equation corresponds to a point on its graph. When these points are plotted, they form a straight line.

Graphing linear equations allows us to see relationships visually, identify patterns, compare equations, and solve many real-world problems.

Learning Targets

• I can graph linear equations in two variables.
• I can identify the slope and intercept of a line.
• I can graph equations from different forms.
• I can determine whether a point lies on a graph.
• I can compare graphs of different linear equations.

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Linear Equations in Two Variables

A linear equation in two variables usually contains:

x

and:

y

Examples include:

y = 3x + 2

2x + y = 8

x − 2y = 6

y = −4x

Although these equations look different, each produces a straight-line graph.

The variables can represent many different quantities, such as:

• Time and distance
• Number of items and total cost
• Hours worked and earnings
• Temperature and time
• Distance and taxi fare

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Points on a Linear Graph

Every point on the graph of an equation represents an ordered pair:

(x, y)

that satisfies the equation.

Consider:

y = 2x + 1

If:

x = 0

then:

y = 2(0) + 1

y = 1

So:

(0, 1)

is on the graph.

If:

x = 2

then:

y = 2(2) + 1

y = 5

So:

(2, 5)

is also on the graph.

There are infinitely many points that satisfy the equation.

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Understanding the Coordinate Plane

A coordinate plane contains two perpendicular axes.

The horizontal axis is the:

x-axis

The vertical axis is the:

y-axis

Their intersection is called the:

origin

with coordinates:

(0, 0)

An ordered pair is written:

(x, y)

Always move along the x-axis first and then vertically according to the y-coordinate.

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Slope-Intercept Form

One of the most useful forms of a linear equation is:

y = mx + b

This is called slope-intercept form.

In this form:

m = slope

and:

b = y-intercept

For example:

y = 3x + 4

has:

Slope = 3

and:

y-intercept = 4

This form makes important information about the graph immediately visible.

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What Is Slope?

The slope describes the steepness and direction of a line.

It tells us how much y changes when x changes.

Slope can be calculated using:

Slope = change in y / change in x

or:

m = rise / run

The rise represents vertical change.

The run represents horizontal change.

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Positive Slope

A line with a positive slope rises from left to right.

For example:

y = 2x + 1

has:

m = 2

This means that whenever x increases by 1, y increases by 2.

We can think of the slope as:

2/1

So from one point, move:

1 unit right

and:

2 units up

to find another point.

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Negative Slope

A line with a negative slope falls from left to right.

For example:

y = −2x + 5

has:

m = −2

This means that when x increases by 1, y decreases by 2.

A negative slope represents a relationship where one quantity decreases as the other increases.

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Zero Slope

A horizontal line has a slope of:

0

For example:

y = 4

Every point on this line has a y-coordinate of 4.

As x changes, y does not change.

Therefore:

change in y = 0

and the slope is zero.

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Undefined Slope

A vertical line has an undefined slope.

For example:

x = 3

Every point has an x-coordinate of 3.

The line is vertical.

Because there is no horizontal change, the calculation for slope would require division by zero, which is undefined.

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Understanding the Y-Intercept

The y-intercept is the point where a line crosses the y-axis.

At every point on the y-axis:

x = 0

Consider:

y = 3x + 5

Set:

x = 0

Then:

y = 3(0) + 5

y = 5

Therefore, the y-intercept is:

(0, 5)

In:

y = mx + b

the value of b gives the y-intercept directly.

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Graphing from Slope-Intercept Form

Consider:

y = 2x + 1

We can identify:

m = 2

and:

b = 1

Start with the y-intercept:

(0, 1)

Then use the slope:

2 = 2/1

Move:

1 right

and:

2 up

This gives another point:

(1, 3)

Repeat the pattern and draw a straight line through the points.

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Worked Example — Positive Slope

Graph:

y = 3x − 2

Identify the slope:

m = 3

Identify the y-intercept:

b = −2

Start at:

(0, −2)

Write the slope as:

3/1

Move:

1 right

and:

3 up

This gives:

(1, 1)

Move again:

1 right

and:

3 up

This gives:

(2, 4)

Draw a straight line through the points.

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Worked Example — Negative Slope

Graph:

y = −1/2 x + 3

The y-intercept is:

(0, 3)

The slope is:

−1/2

This means:

2 right

and:

1 down

Starting from:

(0, 3)

we obtain:

(2, 2)

and then:

(4, 1)

Plot the points and draw a straight line through them.

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Graphing Using a Table of Values

Another method is to create a table.

Consider:

y = 2x − 1

Choose several x-values.

If:

x = −1

then:

y = 2(−1) − 1 = −3

If:

x = 0

then:

y = −1

If:

x = 1

then:

y = 1

If:

x = 2

then:

y = 3

The corresponding points are:

(−1, −3)

(0, −1)

(1, 1)

(2, 3)

Plot the points and connect them with a straight line.

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Why Two Points Are Enough

A straight line is completely determined by two different points.

If you know two points that satisfy a linear equation, you can draw the line through them.

However, when working by hand, finding a third point can be useful as a check.

If the third point does not lie on the same straight line, there may be a calculation or plotting error.

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Graphing from Standard Form

Linear equations may also be written in standard form:

Ax + By = C

For example:

2x + y = 6

One method is to rearrange the equation into slope-intercept form.

Start with:

2x + y = 6

Subtract 2x:

y = −2x + 6

Now we can identify:

Slope = −2

y-intercept = 6

The equation can now be graphed using slope and intercept.

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Worked Example — Rearranging First

Graph:

3x + y = 6

Solve for y:

y = −3x + 6

Therefore:

m = −3

and:

b = 6

Start at:

(0, 6)

Then use:

−3 = −3/1

Move:

1 right

and:

3 down

This produces another point.

Continue the pattern and draw the line.

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Graphing Using Intercepts

Another useful method is to find the x-intercept and y-intercept.

Consider:

2x + 3y = 6

To find the x-intercept, set:

y = 0

Then:

2x = 6

x = 3

So the x-intercept is:

(3, 0)

To find the y-intercept, set:

x = 0

Then:

3y = 6

y = 2

So the y-intercept is:

(0, 2)

Plot these two points and draw the line.

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The X-Intercept

The x-intercept is where a graph crosses the x-axis.

At every point on the x-axis:

y = 0

For example:

y = 2x − 6

Set:

y = 0

Then:

0 = 2x − 6

6 = 2x

x = 3

Therefore, the x-intercept is:

(3, 0)

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Different Forms of Linear Equations

The same line can be represented in different forms.

For example:

y = 2x + 4

can be rearranged to:

2x − y = −4

These equations describe exactly the same line.

Another useful form is point-slope form:

y − y₁ = m(x − x₁)

This form is useful when you know:

• The slope
• One point on the line

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Worked Example — Point-Slope Form

Suppose a line has:

Slope = 2

and passes through:

(3, 4)

Use:

y − y₁ = m(x − x₁)

Substitute:

y − 4 = 2(x − 3)

Expand:

y − 4 = 2x − 6

Add 4:

y = 2x − 2

Now the equation is in slope-intercept form and can be graphed easily.

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Determining Whether a Point Lies on a Graph

A point lies on a graph if its coordinates satisfy the equation.

Consider:

y = 2x + 3

Does:

(2, 7)

lie on the line?

Substitute:

x = 2

and:

y = 7

Then:

7 = 2(2) + 3

7 = 4 + 3

7 = 7

True.

Therefore:

(2, 7) lies on the graph.

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Worked Example — Point Not on a Line

Does:

(3, 5)

lie on:

y = 2x + 1

Substitute:

x = 3

The equation predicts:

y = 2(3) + 1

y = 7

But the point has:

y = 5

Therefore:

(3, 5) does not lie on the graph.

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Comparing Slopes

Consider:

y = 2x + 1

and:

y = 5x + 1

Both have the same y-intercept:

1

But their slopes are different.

The first line has:

m = 2

The second has:

m = 5

The line with slope 5 rises more rapidly.

Therefore, it is steeper.

Slope allows us to compare rates of change between linear relationships.

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Comparing Y-Intercepts

Consider:

y = 3x + 2

and:

y = 3x − 4

Both have slope:

3

But their y-intercepts are different:

2

and:

−4

Because the slopes are equal, the lines have the same steepness and direction.

They are parallel.

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Parallel Lines

Parallel non-vertical lines have the same slope but different y-intercepts.

For example:

y = 2x + 1

and:

y = 2x + 5

Both have:

m = 2

but different intercepts.

They never intersect.

This becomes especially important when studying systems of equations.

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Intersecting Lines

Lines with different slopes generally intersect at one point.

For example:

y = 2x + 1

and:

y = −x + 7

have different slopes.

Their intersection is the point that satisfies both equations.

This connects graphing linear equations directly to solving systems.

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Comparing Linear Equations Visually

Consider:

y = x + 1

y = 2x + 1

y = 3x + 1

All three lines have the same y-intercept:

(0, 1)

Their slopes are:

1, 2, and 3

As the positive slope becomes larger, the line becomes steeper.

Graphing several equations together makes this relationship easy to see.

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Negative Slopes and Steepness

Compare:

y = −x + 4

and:

y = −4x + 4

Both have the same y-intercept.

Their slopes are:

−1

and:

−4

The second line decreases more rapidly.

The magnitude of the slope helps describe steepness.

A slope of −4 is steeper than a slope of −1.

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Linear Graphs in Real Life

Linear graphs are useful whenever one quantity changes at a constant rate compared with another.

Examples include:

• Constant-speed travel
• Hourly wages
• Simple savings plans
• Fixed-price services
• Temperature conversions
• Rental costs
• Phone plans
• Fuel use under simplified conditions

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Worked Example — Taxi Fare

A taxi charges:

$4 starting fee

plus:

$3 per kilometre

Let:

C = cost

and:

d = distance

The equation is:

C = 3d + 4

This has the same structure as:

y = mx + b

The slope:

3

means the cost increases by $3 for each additional kilometre.

The intercept:

4

represents the starting fee.

The graph provides a visual model of the taxi fare.

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Worked Example — Earnings

Suppose a worker earns:

$18 per hour

The equation is:

E = 18h

where:

E = earnings

and:

h = hours worked

The slope is:

18

This represents an earning rate of:

$18 per hour

The intercept is:

0

because before any hours are worked, earnings are $0.

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Worked Example — Savings

Suppose a student already has $50 and saves $20 each week.

The equation is:

S = 20w + 50

where:

S = savings

and:

w = weeks

The slope:

20

represents:

$20 saved per week

The intercept:

50

represents the initial amount.

This shows why slope and intercept can have important real-world meanings.

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Graphs Can Tell a Story

A linear graph contains information about a relationship.

The slope tells us:

How quickly one quantity changes compared with another.

The y-intercept often tells us:

The starting value when x = 0.

For example, in:

C = 5h + 12

the slope may represent:

$5 per hour

while the intercept represents:

a $12 starting fee

Graphing therefore helps us interpret equations rather than simply calculate with them.

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Choosing a Graphing Method

Different equations may be easier to graph using different methods.

For:

y = 3x + 2

using slope and y-intercept is convenient.

For:

2x + 3y = 12

finding the x- and y-intercepts may be convenient.

For an unfamiliar equation, creating a table of values is often reliable.

For an equation given with a known point and slope, point-slope form may be useful.

Choosing an efficient method makes graphing faster and reduces mistakes.

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Checking a Graph

After drawing a graph, you can check it algebraically.

Choose a point that appears to lie on your line.

Substitute its coordinates into the original equation.

If the equation becomes true, the point was plotted correctly.

You can also check:

• Is the y-intercept correct?
• Does the line rise or fall in the correct direction?
• Does its steepness match the slope?
• Are the axes scaled consistently?
• Is the line straight?

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Common Mistakes

Mistake: Reversing x and y

For:

(3, 5)

move to:

x = 3

then:

y = 5

not the other way around.

Mistake: Confusing slope and intercept

In:

y = 4x + 7

the slope is:

4

and the y-intercept is:

7

Mistake: Forgetting the sign of the slope

In:

y = −3x + 2

the slope is:

−3

The line should fall from left to right.

Mistake: Reading a whole number slope incorrectly

A slope of:

3

can be written:

3/1

So move 1 horizontally and 3 vertically.

Mistake: Using inconsistent scales

If one square represents 1 unit on part of an axis, it should not suddenly represent 5 units elsewhere.

Mistake: Assuming a point lies on a line because it looks close

Substitute the coordinates into the equation to verify it exactly.

Mistake: Plotting only one point

One point does not determine a unique line.

Use at least two points.

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Key Terms

Linear equation – an equation whose graph is a straight line.

Coordinate plane – a plane formed by perpendicular x- and y-axes.

Ordered pair – coordinates written as (x, y).

Slope – the rate at which y changes compared with x.

Rise – vertical change.

Run – horizontal change.

Y-intercept – where a graph crosses the y-axis.

X-intercept – where a graph crosses the x-axis.

Slope-intercept form – the form y = mx + b.

Standard form – a form commonly written as Ax + By = C.

Point-slope form – the form y − y₁ = m(x − x₁).

Parallel lines – lines in the same plane that never meet.

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Check Your Understanding

1. Identify the slope and y-intercept of:

y = 4x + 3

2. Identify the slope and y-intercept of:

y = −2x + 5

3. Graph:

y = 2x + 1

using its slope and y-intercept.

4. Graph:

y = −x + 4

using a table of values.

5. Rewrite in slope-intercept form:

2x + y = 8

6. Find the x- and y-intercepts of:

2x + 3y = 12

7. Does:

(2, 7)

lie on:

y = 3x + 1

Explain.

8. Does:

(4, 5)

lie on:

y = 2x − 3

Explain.

9. Compare:

y = 2x + 3

and:

y = 5x + 3

What is the same? What is different?

10. Compare:

y = 4x + 1

and:

y = 4x − 6

What can you say about their graphs?

11. What does a negative slope tell you about the direction of a line?

12. What is the slope of a horizontal line?

13. Why is the slope of a vertical line undefined?

14. A taxi fare is modeled by:

C = 2.5d + 6

Explain the meaning of the slope and intercept.

15. Explain how you can determine whether a point lies exactly on a linear graph without relying only on the picture.

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Key Takeaways

• A linear equation in two variables produces a straight-line graph.
• Every point on the graph represents an ordered pair that satisfies the equation.
• y = mx + b is called slope-intercept form.
• m represents the slope.
• b represents the y-intercept.
• Slope measures the rate of change of y compared with x.
• Positive slopes rise from left to right.
• Negative slopes fall from left to right.
• Horizontal lines have zero slope.
• Vertical lines have undefined slope.
• A line can be graphed using slope and intercept, a table of values, or intercepts.
• Equations in standard form can be rearranged into slope-intercept form.
• A point lies on a graph only if its coordinates satisfy the equation.
• Lines with the same slope and different intercepts are parallel.
• Different slopes produce different rates of change and usually cause lines to intersect.
• The slope and intercept often have meaningful interpretations in real-world situations.
• Choosing an appropriate form and graphing method can make a problem much easier to solve.

Did You Know?

The idea of using coordinates to connect algebra and geometry is strongly associated with René Descartes. The Cartesian coordinate system made it possible to represent equations as geometric shapes. This connection is one of the foundations of modern mathematics, physics, engineering, computer graphics, and data analysis.