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?

y = -3x + 3

and y = ½x – 4

To draw these, what are the gradients and y-intercepts of each?

(line with gradient of -3 and y-intercept of 3)

(line with gradient of ½ and y-intercept of -4)

Plotting these equations

y = -3x + 3

y = ½x - 4

Visually, what represents the solution to these simultaneous equations?

The place where they intersect!

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.

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?

• So there is no solution or we say the two equations are inconsistent.

• They never intersect – they are parallel!

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

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.

Be aware that examiners sometimes try to hide the fact that equations are parallel by multiplying them by a common factor

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?

• So there is an infinite number of solutions or we say the two equations are dependent.

These are the same lines!

That means they intersect in a whole lot of different places.

Did the equations give us a hint?

y = 2x + 1

2y - 4x = 2

Rearranged, we got

y = 2x + 1

2y = 4x + 2This second equation can be divided through by 2 to give

y = 2x + 1

What do you notice?

The equations are exactly the same! (even the constant term)

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)

ActivityActivity: Try solving these different sets of equations on the calculator. What happens?

y = 2x + 1y = 2x + 4

Parallel

y = 2x + 1y = 2x + 1

Same lines

y = -3x + 3y = ½x - 4

Lines intersect

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:

Parallel

No solutions

Match each graph with the description and the number of solutions

Infinite number of solutions

Same lines

Unique solution

Lines intersect

Mix and match 1 Solution:

Parallel

No solutions

Match each graph with the description and the number of solutions

Infinite number of solutionsSame lines

Unique solution

Lines intersect

Mix and match 2:

Parallel

No solutions

Infinite number of solutions

Same lines

Unique solution

Lines intersect

Match each equation with the description and the number of solutions

Mix and match 2 Solution:

ParallelNo solutions

Infinite number of solutionsSame lines

Unique solutionLines intersect

Match each equation with the description and the number of solutions

Mix and match 3:

Parallel

No solutions

Infinite number of solutions

Same lines

Unique solution

Lines intersect

Match each equation with the description and the number of solutionsSome of them might be hidden!

Mix and match 3 Solution:

ParallelNo solutions

Infinite number of solutionsSame lines

Unique solutionLines intersect

Match each equation with the description and the number of solutionsSome of them might be hidden!

Your turnWrite 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