Polkadot JAM Slides - Token2049 - By Dr. Gavin Wood
Linear equations part i
1. LINEAR EQUATIONS PART I
1. Basic Coordinate Plane Info Assignments
2. Review on Plotting Points
3. Finding Slopes
4. x and y intercepts
5. Slope-Intercept Form of a Line
6. Graphing Lines
7. Determine the equation of a line given two points, slope an
2. COORDINATE PLANE
Y-axis
Parts of a plane
1.X-axis
2.Y-axis
QUAD II QUAD I 3.Origin
4.Quadrants I-IV
Origin ( 0 , 0 )
X-axis
QUAD III QUAD IV
3. PLOTTING POINTS
Remember when plotting
points you always start at
B the origin. Next you go left
C
(if x-coordinate is negative)
or right (if x-coordinate is
positive. Then you go up (if
y-coordinate is positive) or
down (if y-coordinate is
negative)
A
D Plot these 4 points
A (3, -4), B (5, 6), C (-4, 5)
and D (-7, -5)
4. SLOPE
Slope is the ratio of the vertical rise to the horizontal
run between any two points on a line. Usually
referred to as the rise over run.Run is 6 Slope triangle between two
because we points. Notice that the slope
went to the triangle can be drawn two
Rise is 10 right different ways.
because we Rise is -10
went up because we
went down
−10 5
The slope in this case is =
Run is -6 −6 3
because we
went to the
left 10 5
The slope in this case is =
6 3
Another way to find slope
5. FORMULA FOR FINDING SLOPE
The formula is used when you know
two points of a line.
They look like A( X 1 , Y1 ) and B ( X 2 , Y2 )
RISE X 2 − X 1 X 1 − X 2
SLOPE = = =
RUN Y2 − Y1 Y1 − Y2
EXAMPLE
6. Find the slope of the line between the two points (-4, 8) and (10, -4)
If it helps label the points. X 1 Y1 X2 Y2
Then use the
formula X 2 − X1 (10) − (−4)
Y2 − Y1 SUBSTITUTE INTO FORMULA (−4) − (8)
(10) − (−4) 10 + 4 14 7
Then Simplify = = =−
(−4) − (8) − 4 + (−8) − 12 6
7. X AND Y INTERCEPTS
The x-intercept is the x-coordinate of a point
where the graph crosses the x-axis.
The y-intercept is the y-coordinate of a point
where the graph crosses the y-axis.
The x-intercept would be 4 and is
located at the point (4, 0).
The y-intercept is 3 and is
located at the point (0, 3).
8. SLOPE-INTERCEPT FORM OF A LINE
The slope intercept form of a line is y = mx + b, where
“m” represents the slope of the line and “b”
represents the y-intercept.
When an equation is in slope-intercept form the
“y” is always on one side by itself. It can not be
more than one y either.
If a line is not in slope-intercept form, then we must
solve for “y” to get it there.
Examples
9. IN SLOPE-INTERCEPT NOT IN SLOPE-INTERCEPT
y = 3x – 5 y – x = 10
y = -2x + 10 2y – 8 = 6x
y = -.5x – 2 y + 4 = 2x
Put y – x = 10 into slope-intercept form
Add x to both sides and would get y = x + 10
Put 2y – 8 = 6x into slope-intercept form.
Add 8 to both sides then divide by 2 and would get y = 3x + 4
Put y + 4 = 2x into slope-intercept form.
Subtract 4 from both sides and would get y = 2x – 4.
10. GRAPHING LINES
BY MAKING A TABLE OR USING THE
SLOPE-INTERCEPT FORM
I could refer to the table method by input-output table or x-y table. For now I
want you to include three values in your table. A negative number, zero, and a
positive number.
Graph y = 3x + 2
INPUT (X) OUTPUT (Y)
-2 -4
0 2
1 5
By making a table it gives me three points, in this case (-2, -4) (0, 2) and (1, 5) to plot
and draw the line.
See the graph.
11. Plot (-2, -4), (0, 2) and (1, 5)
Then draw the line. Make sure your
line covers the graph and has
arrows on both ends. Be sure to
use a ruler.
Slope-intercept graphing
12. Slope-intercept graphing
Steps
1.Make sure the equation is in slope-intercept form.
2.Identify the slope and y-intercept.
3.Plot the y-intercept.
4.From the y-intercept use the slope to get another point to draw the line.
1. y = 3x + 2
2. Slope = 3 (note that this means the
fraction or rise over run could be (3/1)
or (-3/-1). The y-intercept is 2.
3. Plot (0, 2)
4. From the y-intercept, we are going
rise 3 and run 1 since the slope was
3/1.
13. FIND EQUATION OF A LINE GIVEN 2
POINTS
Find the equation of the line between (2, 5) and (-2, -3).
1. Find the slope between the two
points.
1. Slope is 2.
2. Plug in the slope in the slope-
2. y = 2x + b
intercept form.
3. Picked (2, 5) so
3. Pick one of the given points and plug
(5) = 2(2) + b
in numbers for x and y.
4. b = 1
4. Solve and find b.
5. y = 2x + 1
5. Rewrite final form.
Two other ways
14. Steps if given the slope and If given a graph there are three
ways.
a point on the line.
1.Substitute the slope into One way is to find two points on
the slope-intercept form. the line and use the first method
we talked about.
2.Use the point to plug in
for x and y. Another would be to find the
3.Find b. slope and pick a point and use the
second method.
4.Rewrite equation.
The third method would be to find
the slope and y-intercept and plug
it directly into y = mx + b.
