1. Points on a Line
Topic 4.2.1

2. Points on a Line 4.2.1 Topic California Standards:
6.0 Students graph a linear equation and compute the x- and y-intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the region defined by 2x + 6y < 4).
7.0 Students verify that a point lies on a line, given an equation of the line. Students are able to derive linear equations by using the point-slope formula.
What it means for you:
You’ll learn how to show mathematically that points lie on a line.
Key Words:
linear equation
variable
solution set
verify

3. Topic
4.2.1
Points on a Line
You already dealt with lines in Topics and
In this Topic you’ll see a formal definition relating ordered pairs to a line — and you’ll also learn how to show that points lie on a particular line.
(1, 1)
(2, 3)
(3, 5)
(–1, –3)

4. Topic
4.2.1
Points on a Line
Graphs of Linear Equations are Straight Lines
An equation is linear if the variables have an exponent of one and there are no variables multiplied together.
For example:
Linear: 3x + y = 4, x = 6, y = 5 – x
Nonlinear: xy = 12, x2 + 3y = 1, y3 = 20

5. Topic
4.2.1
Points on a Line
Linear equations in two variables, x and y, can be written in the form:
Ax + By = C
The solution set to the equation Ax + By = C consists of all ordered pairs (x, y) that satisfy the equation.
All the points in this solution set lie on a straight line. This straight line is the graph of the equation.

6. Topic
4.2.1
Points on a Line
If the ordered pair (x, y) satisfies the equation Ax + By = C, then the point (x, y) lies on the graph of the equation.
The ordered pair (–2, 5) satisfies the equation 2x + 2y = 6. So the point (–2, 5) lies on the line 2x + 2y = 6.
The point (2, 1) lies on the line 2x + 2y = 6. So the ordered pair (2, 1) satisfies the equation 2x + 2y = 6.

7. Points on a Line 4.2.1 Topic Verifying That Points Lie on a Line
To determine whether a point (x, y) lies on the line of a given equation, you need to find out whether the ordered pair (x, y) satisfies the equation. If it does, the point is on the line.
You do this by substituting x and y into the equation.

8. Topic
4.2.1
Points on a Line
Example 1
a) Show that the point (2, –3) lies on the graph of x – 3y = 11.
b) Determine whether the point (–1, 1) lies on the graph of 2x + 3y = 4.
Solution
a) 2 – 3(–3) = 11
Substitute 2 for x and –3 for y
2 + 9 = 11
11 = 11
A true statement
So the point (2, –3) lies on the graph of x – 3y = 11, since (2, –3) satisfies the equation x – 3y = 11.
Solution continues…
Solution follows…

9. Topic
4.2.1
Points on a Line
Example 1
a) Show that the point (2, –3) lies on the graph of x – 3y = 11.
b) Determine whether the point (–1, 1) lies on the graph of 2x + 3y = 4.
Solution (continued)
b) If (–1, 1) lies on the graph of 2x + 3y = 4, then 2(–1) + 3(1) = 4.
But 2(–1) + 3(1) = –2 + 3
= 1
Since 1 ¹ 4, (–1, 1) does not lie on the graph of 2x + 3y = 4.

10. Points on a Line 4.2.1 Topic Guided Practice
Determine whether or not each point lies on the line of the given equation.
(–1, 2); 2x – y = –4
(3, –4); –2x – 3y = 6
(–3, –1); –5x + 3y = 11
(–7, –3); 2y – 3x = 15
yes: 2(–1) – 2 = –4
yes: –2(3) – 3(–4) = 6
no: –5(–3) + 3(–1) = –18 ¹ 11
yes: 2(–3) – 3(–7) = 15
Solution follows…

11. Points on a Line 4.2.1 Topic Guided Practice
Determine whether or not each point lies on the line of the given equation.
(–2, –2); y = 3x + 4
(–5, –3); –y + 2x = –7
(–2, –1); 8x – 15y = 3
yes: –2 = 3(–2) + 4
yes: –(–3) + 2(–5) = –7
no: 8(–2) – 15(–1) = –1 ¹ 3
Solution follows…

12. Points on a Line 4.2.1 Topic Guided Practice
Determine whether or not each point lies on the line of the given equation.
(1, 4); 4y – 12x = 3
( , – ); 6x – 16y = 7
( , – ); –3x – 10y = 2
no: 4(4) – 12(1) = 4 ¹ 3 1 3 1 4
no: 6( ) – 16(– ) = 6 ¹ 7 1 3 4 2 3 2 5
yes: –3( ) – 10(– ) = 2 2 3 5
Solution follows…

13. Points on a Line 4.2.1 Topic Independent Practice
In Exercises 1–4, determine whether or not each point lies on the graph of 5x – 4y = 20.
(0, 4)
(4, 0)
(2, –3)
(8, 5)
no: 5(0) – 4(4) = –16 ¹ 20
yes: 5(4) – 4(0) = 20
no: 5(2) – 4(–3) = 22 ¹ 20
yes: 5(8) – 4(5) = 20
Solution follows…

14. Points on a Line 4.2.1 Topic Independent Practice
In Exercises 5–8, determine whether or not each point lies on the graph of 6x + 3y = 15.
(2, 1)
(0, 5)
(–1, 6)
(3, –1)
yes: 6(2) + 3(1) = 15
yes: 6(0) + 3(5) = 15
no: 6(–1) + 3(6) = 12 ¹ 15
yes: 6(3) + 3(–1) = 15
Solution follows…

15. Points on a Line 4.2.1 Topic Independent Practice
In Exercises 9–12, determine whether or not each point lies on the graph of 6x – 6y = 24.
(4, 0)
(1, –3)
(100, 96)
(–400, –404)
yes: 6(4) – 6(0) = 24
yes: 6(1) – 6(–3) = 24
yes: 6(100) – 6(96) = 24
yes: 6(–400) – 6(–404) = 24
Solution follows…

16. Points on a Line 4.2.1 Topic Independent Practice
13. Explain in words why (2, 31) is a point on the line x = 2, but not a point on the line y = 2.
x = 2 includes all points whose x-coordinate is 2, so (2, 31) is a point on x = 2; the y-coordinate is 31, so (2, 31) does not lie on the line y = 2.
14. Determine whether the point (3, 4) lies on the lines 4x + 6y = 36 and 8x – 7y = 30.
4x + 6y = 36: 4(3) + 6(4) = 36
8x – 7y = 30: 8(3) – 7(4) = –4 ¹ 36
(3, 4) does lie on the line 4x + 6y = 36, but not on the line 8x – 7y = 30, so it doesn’t lie on both lines.
Solution follows…

17. Points on a Line 4.2.1 Topic Round Up
You can always substitute x and y into the equation to prove whether a coordinate pair lies on a line.
That’s because if the coordinate pair lies on the line then it’s actually a solution of the equation.