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Transcript
Algebra 1 Notes SOL A.6 (4.5) Slope-Intercept Form
Mrs. Grieser
Name: _______________________________ Block: _______ Date: _____________
Slope-Intercept Form
We’ve looked at linear equations in standard form: Ax + By = C
Let’s solve the standard form for y:
Ax + By = C
By = C – Ax
C Ax
y= B
B
Given
Subtract Ax from both sides (subtraction property of equality)
Divide by B on both sides (division property of equality)
Let’s introduce new constants: Let b =
C
A
and Let m = - .
B
B
C Ax
= b + mx
B
B
Using the commutative property of addition, we get…y = mx + b
Then y =
Since we derived y = mx + b from the standard form of a linear equation, it, too, describes a
linear equation. We call y = mx + b the slope-intercept form of linear equations. As we saw
from our experiment with lines in this form, m is the slope of the line and b is the y-intercept.
Slope-Intercept Form
y = mx + b, where m = slope and b = y-intercept
Example: Identify the slope and y-intercept of a linear equation.
a) y = 3x + 4
b) 3x + y = 2
Compare to y = mx + b
Put in slope-intercept form by solving for y.
m = 3 and b = 4
c) 3x – 3y = 12 (you try)
y = -3x + 2
m = -3 and b = 2
Graphing Using Slope-Intercept Form
Given a linear equation, we can graph it by following the steps below:
1. Put linear equation in slope-intercept form.
2. Identify slope (in fraction form) and y-intercept.
3. Plot the y-intercept.
4. Use the slope to locate a second point on the line (by starting at the y-intercept, and then
going up or down the number of units in the slope’s numerator and left or right the number
of units in its denominator).
5. Connect the two points.
1
Algebra 1 Notes SOL A.6 (4.5) Slope-Intercept Form
Mrs. Grieser
Example: Graph 2x + y = 3
1. Put linear equation in slope-intercept form.
2x + y = 3 ⇒ y = -2x + 3
2. Identify slope and y-intercept.
2
m = -2 = - , b = 3
1
3. Plot the y-intercept.
Plot point (0, 3)
4. Use the slope to locate a second point on the line.
Starting at point (0, 3), go down -2 and over to the right 1. This is your second point (1, 1).
5. Connect the two points.
Parallel Lines
Remember solving equations with variables on both sides? We discussed that each side
represents a linear equation. The point of intersection was the solution to the equation.
Sometimes we had “no solution.” This was the case where the lines were parallel, as parallel
lines never intersect.
Example: Solve 2x + 4 = 2x - 4
The figure to the right shows the graph of each side of the equation.
We find that the above equation has no solution because the lines are
parallel. We also see that the two lines have the same slope.
Conclusion: Two lines with the same slope are parallel.
Example: Determining parallel lines
Which lines in the figure to the right are parallel?
Solution: Determine the slope of each line. The lines with the same
slope are parallel.
Line a: m =
0 − ( −1) 1
= ;
2 − ( −1) 3
Line b: m =
- 1 − ( −3) 2
= ;
5
5 − (0)
Line c: m =
1
- 3 − ( −5) 2
=
=
6 3
4 − (-2)
Conclusion: Since line a and line c have the same slope, they are parallel.
You try: Are the two lines parallel?
a) y = 5x – 7 and 5x + y = 7
b) y = 3x + 2 and -7 + 3x = y
2
1
c) y = - x and x + 2y = 18
2