C. Algebra · Linear and Non-Linear Relations

Graphing x = k, y = k, x + y = k and x − y = k

C4.2 A1About 65 minutes

Learning goals

I can graph vertical lines x = k and horizontal lines y = k, and explain why they look that way.
I can graph x + y = k and x − y = k using intercepts, and describe their slopes.
I can find the point where two of these lines intersect and verify it.
I can use these equations to describe real situations, such as two numbers with a fixed sum or difference.

Words to know

constant
A fixed number. In the equations x = k and x + y = k, the letter k stands for a constant, like 3 or −2.

vertical line
A line going straight up and down. Every point has the same x-coordinate: x = k.

horizontal line
A line going straight across. Every point has the same y-coordinate: y = k.

point of intersection
The point where two lines cross. Its coordinates make both equations true.

Think of a pair

Think of two numbers that add to 6. Maybe 2 and 4, or 6 and 0, or 7 and −1, or 2.5 and 3.5. There are infinitely many pairs. If you plot every pair (x, y) with x + y = 6, they make a straight line.

In this lesson you will graph four simple kinds of equations. Each one describes a rule that every point on its graph follows.

x = k and y = k

x = 3 means "every point has x-coordinate 3, and y can be anything": (3, −2), (3, 0), (3, 1), (3, 5) ... These points stack up into a vertical line crossing the x-axis at 3. Its slope is undefined.

y = −2 means "every point has y-coordinate −2, and x can be anything". The points line up horizontally, crossing the y-axis at −2. Its slope is 0.

Two such lines always meet at exactly one point: x = 3 and y = −2 meet at (3, −2).

The vertical line x = 3 (solid) and the horizontal line y = −2 (dashed) cross at (3, −2). The small dots (3, 1) and (3, 3) are two more points on x = 3: the x-coordinate is always 3.

x + y = k and x − y = k

Rearrange to see the slope:

x + y = k becomes y = −x + k: slope −1, y-intercept k. The x-intercept is also k: (k, 0).
x − y = k becomes y = x − k: slope 1, y-intercept −k, x-intercept k.

The quickest way to graph them is the intercept method: set x = 0 to find the y-intercept, set y = 0 to find the x-intercept, plot both points and join them.

Left: x + y = −2, x + y = 2 and x + y = 6 are parallel lines with slope −1; the dots show the intercepts (6, 0) and (0, 6) of x + y = 6. Right: x − y = 2 (solid) and x − y = −2 (dashed) are parallel with slope 1; the dots show the intercepts (2, 0) and (0, −2) of x − y = 2.

Equation	Graph	Slope	x-intercept	y-intercept
x = k	Vertical line	Undefined	k	None (unless k = 0, then it is the y-axis)
y = k	Horizontal line	0	None (unless k = 0, then it is the x-axis)	k
x + y = k	Falls to the right	−1	k	k
x − y = k	Rises to the right	1	k	−k

Characteristics of the four kinds of lines.

Worked example

Example 1: Graph x + y = 6 and x − y = 2 and find where they meet

Graph both lines on one grid. Find the point of intersection and verify it.

x + y = 6: x = 0 gives y = 6, so (0, 6). y = 0 gives x = 6, so (6, 0). Join them.
x − y = 2: x = 0 gives −y = 2, so y = −2: (0, −2). y = 0 gives x = 2: (2, 0). Join them.
The lines cross at (4, 2).
Verify in both equations: 4 + 2 = 6 ✓ and 4 − 2 = 2 ✓.

The point of intersection is (4, 2): the only pair of numbers with a sum of 6 and a difference of 2.

x + y = 6 (solid) and x − y = 2 (dashed) meet at (4, 2). The point of intersection is the one point whose coordinates make both equations true.

Worked example

Example 2: A fence with a fixed perimeter

Amira has 20 m of fencing for a rectangular garden. Write an equation relating the length l and the width w, graph it, and describe which part of the line makes sense.

Perimeter: 2l + 2w = 20. Divide by 2: l + w = 10. This is the form x + y = k with k = 10.
Intercepts: (10, 0) and (0, 10). The graph is a line with slope −1: every extra metre of length takes a metre away from the width.
Lengths and widths must be positive, so only the part of the line between the intercepts makes sense, not including the intercepts themselves (a garden 10 m long and 0 m wide is not a garden).

l + w = 10, the segment between (0, 10) and (10, 0), not including its endpoints. For example, 6 m by 4 m or 7.5 m by 2.5 m.

Worked example

Example 3: A fixed difference

Noah's sister is 3 years older than Noah. Let x be the sister's age and y be Noah's age. Write and describe the relation.

The difference is always 3: x − y = 3, or y = x − 3.
Slope 1: each year, both ages go up by 1.
When the sister is 15, Noah is 15 − 3 = 12, so (15, 12) is on the line.

x − y = 3, a line with slope 1. In context only positive ages make sense.

Just for fun

The lines x + y = 10 and x − y = 10 went to a party. They didn't have much in common: only the point (10, 0).

Hands-on activity

Floor-grid lines

You need: a floor with square tiles (or a grid drawn with sidewalk chalk outside, or masking tape on the floor), a few small objects as markers (coins, buttons, LEGO bricks), family members if available.

Choose one tile corner as the origin. Decide which direction is the positive x-axis and which is the positive y-axis.
Place markers on five points where x + y = 4, including some with negative coordinates (like (5, −1)). Do the markers line up?
Using a different kind of marker, place five points where x − y = 2. Where do your two lines meet? Stand on that spot.
Now mark x = −1. Which points on it are also on x + y = 4?

What to look for: the markers for each rule form a straight line. x + y = 4 and x − y = 2 meet at (3, 1). x = −1 and x + y = 4 meet at (−1, 5).

Your turn

Describe the graph of x = −4.

Describe the graph of y = 5.

Find the intercepts of x + y = 8.

Find the intercepts of x − y = 5.

Is (7, −2) on x + y = 5? Is it on x − y = 5?

Where do x = 2 and y = 7 intersect?

Find the point of intersection of x + y = 10 and x − y = 4. Verify it.

More practice: warm-up

What is the slope of y = −3?

What is the slope of x = 6?

Name three points on x + y = 4.

What is the slope of every line of the form x + y = k?

What is the slope of every line of the form x − y = k?

More practice: on your own

Find the intercepts of x + y = −3 and describe the line.

Rewrite x − y = −1 in the form y = mx + b and give its intercepts.

Where does x + y = 7 meet y = 2?

A rectangular pen has a perimeter of 16 m. Write an equation for length l and width w. Why does only part of the graph make sense?

A sister is 4 years older than her brother. Let x be her age and y his age. Write the equation and find the point on the line when the brother is 10.

Challenge

Find k so that x + y = k passes through (3, −5). Then find k so that x − y = k passes through the same point.

The lines x + y = 6, x − y = 2 and x = 0 form a triangle. Find its three vertices and its area.

Explain in two different ways why x + y = 3 and x + y = 7 never intersect.

Check your understanding

Quiz

What does the graph of x = −2 look like?

What is the slope of x + y = 9?

What is the y-intercept of x − y = 4?

Where do x = 5 and y = −1 intersect?

Where do x + y = 8 and x − y = 2 intersect?

Which lines are parallel?

For parents and teachers

Big idea: Expectation C4.2 asks students to graph relations of the forms x = k, y = k, x + y = k, x − y = k, ax + by = k and xy = k (and the related inequalities) and identify their characteristics and the points they define. This lesson covers the first four; the next lessons cover ax + by = k, xy = k and inequalities.

Common difficulties:

Mixing up x = k and y = k. x = 3 is vertical (it crosses the x-axis at 3). Ask: "Which coordinate stays the same?"
Sign errors in x − y = k, especially for the y-intercept (−k, not k).
Not verifying a point of intersection in both equations.

How to help: The floor-grid activity is a fun, physical way to see these lines. Puzzles like "two numbers add to 10 and differ by 4" connect the algebra to mental math.