Introduction to Linear Systems
Learning Goals
By the end of the lesson I will be able to:
Remember:
Graphing a line in y = mx + b mode:
b → represents a point: (0, b), the y-intercept
y = 2x - 2
m → represents the slope: This allows us to find other points
rise → The vertical difference
run → The horizontal difference
Remember:
Graphing a line in standard form:
Use substitution to find points!
3x - 2y + 6 = 0
Pick any value: x = ______
Now replace every x with ______
Solve for y
This gives us a point: ( , )
Remember
We need to do this twice! Two points are necessary!
3x - 2y + 6 = 0
Another value for x = ______
Replace every x with ______
Solve for y
Now we have our second point: ( , )
Remember
Plot the 2 points:
( , ) and
( , )
Linear Systems
What happens when we have two lines on the same graph?
This is called a system of linear equations.
These two lines meet at one point, called the point of intersection.
This point is the solution to the system.
Linear Systems
What else can we see:
Identify the slopes of the two lines?
A: B:
Different slopes mean we have ONE point of intersection.
A
B
Linear Systems
What are the slopes of these two lines:
A: B:
How many points of intersection are there? _____
If the lines have the same slope, they are parallel and have 0 points of intersection.
A
B
Identify
State whether each line is parallel or intersecting to the line y = 4x + 1
a) y = -4x + 1 d) y = x + 1
b) y = 4x + 7
c) y = 2x + 1 e) y = - x
1
4
1
4
Bring it all together
A Systems of Linear Equations:
A Point of Intersection:
Homework
Graph the following systems of linear equations on the same graph. Identify:
Homework
-3x + 4 y = 3x - 5 y = 2x + 2
2x - 1 y = -x - 1 y = x - 1
x - y = -1 x - 2y = -1 Make one of
3x + y = 9 3x + 5y = 8 your own