Download Solving Systems by Elimination – An Investigation

Math 11 Academic
Mathematician: ______________________
Solving Systems by Elimination – An Investigation
Purpose: You will be using Autograph software to examine a new method for finding a point of intersection of two lines.
1. We will be working with two linear equations:
L1 :
5x – 2y + 12 = 0
L2 :
x + 4y – 2 = 0
a) How do you know these equations are linear ?_________________________
_________________________________________________________________
b) Graph these lines on the same grid in Autograph.
c) Write the coordinates of the point of intersection of L1 and L2: ( ____, ____ )
2. a) Multiply L1 by 2. State the resulting equation: _________________________
b) Graph this equation. Describe the graph: ____________________________
c) Explain: _______________________________________________________
3. a) Multiply L2 by –3. State the resulting equation: ________________________
b)
Graphing Using Autograph
Open Autograph (double click on
the icon)
Select Axes and Edit Axes.
Select Ranges:
Change the settings:
X min = -10
X max = 10
Y min = -10
Y max = 10
Select Options:
Click Square Format
Click on OK
Click on Equation and Enter
Equation
Type your equation and click OK.
Predict what the graph would look like: __________________________________________________.
c) Verify your predictions by graphing L2 in Autograph.
Based on your results from #2 and #3, complete the following statement:
When you multiply the equation of a straight line by a constant,
4. a)
Add L1 and L2 :
L1 :
5x – 2y + 12 = 0
L2 :
x + 4y – 2 = 0
Graph this line on
the same grid.
6x + 2y + 10 = 0
b) Is your new graph the same as your original graphs? ________
Does it have any points in common with L1 and L2 ? ________
If so, what are they? ____________________
5. a) Write the equation of
b) Write the equation of
(2) × L1 : ____________________
(-3) × L2 : ____________________
Graph this line on
the same grid.
ADD: ____________________
c) Does this line have any points in common with the original lines? ________
If so, what are they? _____________________
6. a) Complete:
5x – 2y + 12 = 0
3
____________________
x + 4y – 2 = 0
5
____________________
ADD :
b) Does this line also pass through (-2,1)? _____
____________________
Graph this line
on the same
grid.
7. a) Choose two numbers: _____ and _____
Multiply L1 by the first number:
5x – 2y + 12 = 0
Multiply L2 by the second number:
x + 4y – 2 = 0
___________________
___________________
ADD:
___________________
b) Use Autograph to graph your equation on the same grid.
Explain what you notice: ________________________________________________________________________
c) Compare your results with other students. Use your observations to complete the following statement:
When you multiply two linear equations by different numbers and add the resulting equations,
We can use this to create equations with only one variable!
Recall that adding opposites gives zero.
For example, 3 + (-3) = 0 and -6x + 6x =0
What if we choose our multipliers so that the coefficients of x are
opposites?
5x – 2y + 12 = 0
x + 4y – 2 = 0
5x – 2y + 12 = 0
  5
–5x – 20y + 10 = 0
Graph this line
on the same grid.
ADD: ________________
and solve:
Describe this line:____________________________________
*Notice that in our new equation we have eliminated x.
On the other hand, what if we choose our multipliers so that the coefficients of y are
opposites?
5x – 2y + 12 = 0
 2
10x – 4y + 24 = 0
x + 4y – 2 = 0
x + 4y – 2 = 0
Can you see what
the point of
intersection is
without graphing?
Graph this line on
the same grid.
ADD: ________________
and solve:
Describe this line: _______________________________
*In this new equation we have eliminated y.
Describe how you can use what you learned to find a point of intersection of two linear equations without graphing:
________________________________________________________________________________________________