Section 3.6—Lines in the coordinate plane
Ali Landers 92
Quick vocab:
Slope-intercept form: y=mx+b
Standard form of a linear equation: Ax+By=C
Point-slope form: y-y1=m(x-x1)
Y-intercept: the y coordinate of the point where a line crosses the y-axis
X-intercept: the x coordinate of the point where a line crosses the x-axis
Also, the slope of a horizontal line is zero and the slope of a vertical line is undefined.
Graphing lines using intercepts
Graph for 2x+2y=12.
Step 1 To find the y-intercept, substitute 0 for x. Solve for y.
2x+2y=12
2(0)+2y=12
2y=12
y=6
The y-intercept is 6. A point on the line is (0,6).
Step 2 To find the x-intercept, substitute 0 for y. Solve for x.
2x+2y=12
2x+2(0)=12
2x=12
x=6The x-intercept is 6.
A point on the line is (6,0).
Step 3 Plot (0,6) and (6,0). Draw the line containing the two points.
Transforming to slope-intercept form
Graph the equation -10x-2y=4
Step 1 Transform the equation toslope-intercept form.
-10x+2y=(-4)
2y=10x-4
2y=10x-4
2 2
y=5x-2
The y-intercept is -2 and the slope is 5.
Step 2 Use the slope and y-intercept to plot two points and draw the line containing them.
Using point-slope form
Write the equation of the line through point P (-1,4) with a slope of 3.
y-y1=m(x-x1)Use point-slope form.
y-4=3[x-(-1)]Substitute 3 for m and (-1,4) for (x1, y1)
y-4=3(x+1)Simplify.
Writing the equation of a line given two points
Write the equation of the line through A (-2,3) and B (1,-1).
Step 1 Find the slope
M=y2-y1
x2-x1
m= -1-3
1-(-2) Substitute (-2,3) for (x1, y1) and (1,-1) for (x2,y2).
m= -4
3 Simplify.
Step 2Select one of the points. Write an equation in point-slope form.
y-y1=m(x-x1)
y-3= -4/3[x-(-2)]Substitute (-2,3) for (x1, y1) and -4/3 for the slope.
y-3= -4/3(x+2)Simplify.
Equations of horizontal and vertical lines
Every point on the horizontal line through point (3, -2) has a y-coordinate of -2. The equation of the line is y= -2.
The line crosses the y-axis at (0, -2).
Every point on the vertical line through point (3, -2) has an x-coordinate of 3. The equation of the line is x=3.
The line crosses the x-axis at (3,0).
Practice Problems
- Graph -2x+4y= -8
- Graph -5x+y= -3
- Write an equation of the line with slope -1 that contains the point P (2, -4).
- Graph y= -½x -2
- Write an equation of the line that contains the point P (5,0) and point Q (7, -3).
- Write equations of the horizontal and vertical lines that contain the point (5, -1).
- Graph y=-3x -5.