Transcript File

What is involved for achieved?
Forming and solving 3 simultaneous
equations from words.
Giving your solution back in context.
What is involved for merit?
Identify the type of solution you have
Equations with 2 variables can be drawn as
lines in geometric space.
For example, consider the following 2 x 2 simultaneous
equations:
3x + y = 3
x – 2y = 8
If we rearrange these to the general form of the equation of a
straight line y = mx + c what do we get?
and
y = -3x + 3
y = ½x – 4
(line with gradient of -3 and y-intercept of 3)
(line with gradient of ½ and y-intercept of -4)
To draw these, what are the gradients and y-intercepts of each?
Plotting these equations
Visually, what represents the solution
to these simultaneous equations?
The place where they intersect!
y = -3x + 3
Plotting the above equations we see
they intersect at a single point (2, -3).
In this case x = 2 and y = -3
So we have a unique solution.
y = ½x - 4
Graph these lines and find the
place where they intersect
Rearrange these to the general form of the
equation of a straight line y = mx + c and
graph them
y = 2x + 1
y – 2x = 4
What happened?
• They never intersect – they are parallel!
• So there is no solution or we say the two
equations are inconsistent.
Did the equations give us a
hint?
y = 2x + 1
y – 2x = 4
Rearranged, we got
y = 2x + 1
y = 2x + 4
What do you notice?
The equations are
the same, except for
the y-intercept (or the
constant term)
This means they are
parallel
Would these lines be parallel?
y = 2x + 1
3y = 6x + 9
Be aware that examiners sometimes
try to hide the fact that equations are
parallel by multiplying them by a
common factor
Well, lets put them in the same form of y = mx + c first.
So divide the second equation by 3 to get:
y = 2x + 3
Is this parallel to the first equation?
Yes. Same equation except for the constant.
One more type to check
Rearrange these to the general form of the
equation of a straight line y = mx + c and
graph them
y = 2x + 1
2y - 4x = 2
What happened?
These are the same lines!
That means they intersect
in a whole lot of different
places.
• So there is an infinite number of solutions or
we say the two equations are dependent.
Did the equations give us a
hint?
y = 2x + 1
2y - 4x = 2
Rearranged, we got
y = 2x + 1
2y = 4x + 2
The equations are
exactly the same!
(even the constant
term)
This second equation can be divided through by 2 to give
y = 2x + 1
What do you notice?
Examiners sometimes
try to hide this by
multiplying through by a
common factor
In Summary
• There are 3 solutions:
- One unique solution (lines intersect)
- No solutions (lines are parallel)
- Many solutions (lines are the same)
Activity
Activity: Try solving these different sets of
equations on the calculator. What happens?
y = -3x + 3 Lines
y = ½x - 4 intersect
y = 2x + 1
Same lines
y = 2x + 1
y = 2x + 1
Parallel
y = 2x + 4
How do you know which set
of equations is which type?
• The only one that the calculator will solve is the
one with the unique solution.
• So the calculator will tell you if it is unique, but if
it is not unique, you need to know what to look
for to figure out what type of solution it is.
Mix and match 1:
Match each graph with the description and the number of solutions
Parallel
Lines intersect
Same lines
No solutions
Infinite number of solutions
Unique solution
Mix and match 1 Solution:
Match each graph with the description and the number of solutions
Parallel
No solutions
Lines intersect
Unique solution
Same lines
Infinite number of solutions
Mix and match 2:
Match each equation with the description and the number of solutions
Parallel
No solutions
Same lines
Lines intersect
Infinite number of solutions
Unique solution
Mix and match 2 Solution:
Match each equation with the description and the number of solutions
Parallel
No solutions
Same lines
Infinite number of solutions
Lines intersect
Unique solution
Mix and match 3:
Match each equation with the description and the number of solutions
Some of them might be hidden!
Infinite number of solutions
Same lines
Lines intersect
No solutions
Parallel
Unique solution
Mix and match 3 Solution:
Match each equation with the description and the number of solutions
Some of them might be hidden!
Parallel
No solutions
Same lines
Infinite number of solutions
Lines intersect
Unique solution
Your turn
Write a system of two linear equations with two
variables to represent the following geometric
situations:
Two lines that intersect at a point (unique)
Two lines that are parallel (inconsistent)
Two lines that are the same (dependent)
Do the same thing for all of the above, but be
sneaky and try to hide what you are doing (multiply
through by constants)
Choose one of these to give to a partner and see if
they can figure out what type of solution it is