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INTERCEPTS AND SLOPES OF STRAIGHT LINES
♦ To find the x- and y-intercepts of a straight line
Example 1
Find the x-intercept and the y-intercept of the graph of the equation 2x + 3y = 6
The graph of the equation 2x + 3y = 6 is shown below. The graph crosses the x-axis at the point
(3, 0). This point is called the x-intercept. The graph also crosses the y-axis at the point (0, 2). This
point is called the y-intercept.
To find the x-intercept , let y = 0. To find the y-intercept , let x = 0.
(Any point on the x-axis has y-coordinate 0) (Any point on the y-axis has x-coordinate 0)
2x + 3y = 6 2x + 3y = 6
2x + 3(0) = 6 2(0) + 3y = 6
2x = 6 3y = 6
x = 3 y = 2
The x-intercept is (3, 0) The y-intercept is (0, 2)
For any equation of the form y = mx + b , the y-intercept is (0, b)
Example 2
Find the y-intercept of y = 3x + 4
y = 3x + 4 = 3(0) + 4 = 4 (let x = 0)
The y-intercept is (0, 4)
- To find the slope of a straight line
Slope Formula
If P 1 (x 1 , y 1 ) and P 2 (x 2 , y 2 ) are two points on a line and
y 2 – y
x 1 ≠ x 2 , then m = x2 – x
If x 1 = x 2 , the slope is undefined.
Example 3
Find the slope of the line containing the points (-1, 1) and (2, 3)
Let P 1 be (-1, 1) and P 2 be (2, 3)
Then,
X 1 = -1, Y 1 = 1, X 2 = 2, Y 2 = 3.
2 1
2 1
− −
X X
Y Y
m
The slope is 3
A line that slants upward to the right always has a positive slope.
Example 6
Find the slope of the line containing the points (2, -2) and (2, 4).
Let P1 be (2, -2) and P2 be (2, 4).
2 1
2 1
−
X X
Y Y
m
The slope is undefined.
- To graph a line using the slope and the y-intercept
Slope-Intercept Equation of a line
An equation of the form y = mx + b is called slope-intercept form of a straight line. The slope of the
line is m , the coefficient of x. The y-intercept is (0, b), where b is the constant term of the equation.
The graph of the equation 1
3
y = x + is shown
at the right. The points (-3, -1) and (3, 3) are on
the graph. The slope of the line between the two
points is 3
m =.
Observe that the slope of the line is the coefficient of
x in the equation 1 3
y = x +. Also recall that the
y-intercept is (0, 1), where 1 is the constant term of
the equation.
The following equations are written in slope-intercept form.
y = 2x – 3 Slope = 2; y-intercept = (0, -3)
y = -x + 2 Slope = -1 (-x = -1 x); y-intercept = (0, 2)
x y = Slope = (^)
x
x
=^ ; y-intercept = (0, 0)
Example 7
Graph y = 2x – 3
y-intercept = (0, b) = (0, -3)
m 2
Change in y
Change in x
Beginning at the y-intercept, move right 1 unit
(change in x) and then up 2 units (change in y).
(1, -1) is a second point on the graph.
Draw a line through the two points (0, -3)
and (1, -1).
