Transcript Points on a Line

Points on a Line
Topic 4.2.1
1
Topic
4.2.1
Points on a Line
California Standards:
What it means for you:
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).
You’ll learn how to show
mathematically that points
lie on a line.
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.
•
•
•
•
Key Words:
linear equation
variable
solution set
verify
2
Topic
4.2.1
Points on a Line
You already dealt with lines in Topics 4.1.3 and 4.1.4.
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.
(3, 5)
(2, 3)
(1, 1)
(–1, –3)
3
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,
Nonlinear: xy = 12,
2x = 6,
y=5–x
x2 + 3y = 1,
8y3 = 20
4
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.
5
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.
6
Topic
4.2.1
Points on a Line
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.
7
Topic
Points on a Line
4.2.1
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.
8
Solution
Solution
continues…
follows…
Topic
4.2.1
Example
Points on a Line
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  4, (–1, 1) does not lie on the graph
of 2x + 3y = 4.
9
Topic
4.2.1
Points on a Line
Guided Practice
Determine whether or not each point lies on the line of
the given equation.
1. (–1, 2);
2x – y = –4
yes: 2(–1) – 2 = –4
2. (3, –4);
–2x – 3y = 6
yes: –2(3) – 3(–4) = 6
3. (–3, –1);
–5x + 3y = 11
no: –5(–3) + 3(–1) = –18  11
4. (–7, –3);
2y – 3x = 15
yes: 2(–3) – 3(–7) = 15
10
Solution follows…
Topic
4.2.1
Points on a Line
Guided Practice
Determine whether or not each point lies on the line of
the given equation.
5. (–2, –2);
y = 3x + 4
yes: –2 = 3(–2) + 4
6. (–5, –3);
–y + 2x = –7
yes: –(–3) + 2(–5) = –7
7. (–2, –1);
8x – 15y = 3
no: 8(–2) – 15(–1) = –1  3
11
Solution follows…
Topic
Points on a Line
4.2.1
Guided Practice
Determine whether or not each point lies on the line of
the given equation.
8.
(1, 4);
1
1
9. ( , – );
3
4
2
2
10. ( , – );
3
5
4y – 12x = 3
no: 4(4) – 12(1) = 4  3
6x – 16y = 7
1
1
no: 6( ) – 16(– ) = 6  7
3
4
–3x – 10y = 2
2
2
yes: –3( ) – 10(– ) = 2
3
5
12
Solution follows…
Topic
4.2.1
Points on a Line
Independent Practice
In Exercises 1–4, determine whether or not each point lies
on the graph of 5x – 4y = 20.
1. (0, 4)
no: 5(0) – 4(4) = –16  20
2. (4, 0)
yes: 5(4) – 4(0) = 20
3. (2, –3)
no: 5(2) – 4(–3) = 22  20
4. (8, 5)
yes: 5(8) – 4(5) = 20
13
Solution follows…
Topic
4.2.1
Points on a Line
Independent Practice
In Exercises 5–8, determine whether or not each point lies
on the graph of 6x + 3y = 15.
5. (2, 1)
yes: 6(2) + 3(1) = 15
6. (0, 5)
yes: 6(0) + 3(5) = 15
7. (–1, 6)
no: 6(–1) + 3(6) = 12  15
8. (3, –1)
yes: 6(3) + 3(–1) = 15
14
Solution follows…
Topic
4.2.1
Points on a Line
Independent Practice
In Exercises 9–12, determine whether or not each point
lies on the graph of 6x – 6y = 24.
9. (4, 0)
yes: 6(4) – 6(0) = 24
10. (1, –3)
yes: 6(1) – 6(–3) = 24
11. (100, 96)
yes: 6(100) – 6(96) = 24
12. (–400, –404)
yes: 6(–400) – 6(–404) = 24
15
Solution follows…
Topic
4.2.1
Points on a Line
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  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.
16
Solution follows…
Topic
4.2.1
Points on a Line
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.
17