SOLVING SIMULTANEOUS EQUATIONS

GRAPHICALLYSlideshow 32, Mathematics

Mr Richard Sasaki, Room 307

OBJECTIVES

• Consider the two methods learned previously regarding how to draw graphs

•Understand how to find unique solutions for simultaneous equations represented graphically

DRAWING LINES

If we are given an equation in any format (let’s say as we’re dealing with simultaneous equations, do draw a line we can either…

• Change the subject

• Find two pairs of co-ordinates

Both are necessary so we will practice using both. Let’s start with changing the subject.

SIMULTANEOUS EQUATIONS – CHANGING

THE SUBJECTLet’s try an example.

Graphically, solve the simultaneous equations and .

Firstly, let’s rearrange both.𝑥+𝑦=8⟹𝑦=−𝑥+8𝑥− 𝑦=2⟹𝑦=𝑥−2Next, let’s draw these lines!

SIMULTANEOUS EQUATIONS – CHANGING

THE SUBJECT𝑥+𝑦=8⟹𝑦=−𝑥+8𝑥− 𝑦=2⟹𝑦=𝑥−2

First, let’s draw and label .

𝑥+𝑦=8

And next .

𝑥− 𝑦=2 What do we do next?

Right! Find the co-ordinates where the lines cross…We get (5, 3), so…

𝑥=5 , 𝑦=3

(5 ,3)

ANSWERS

1.𝑥=1. 4̇ , 𝑦=−2 ˙.22. 𝑥=2 , 𝑦=63. 𝑥=2 , 𝑦=34.𝑥=3 , 𝑦=05. 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠

6.𝑛𝑜𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠

HOW MANY SOLUTIONS?

Hopefully you found out…

When two lines cross…

We get one solution.

When two lines are parallel…

We get no solutions.

When two lines are on top of each other…

We get infinite solutions!

SOLVING SIMULTANEOUS EQUATIONS GRAPHICALLY

One problem with solving simultaneous equations graphically…

How do we solve this?? Or this??Our pencils and eyes aren’t accurate enough sometimes. So we’d need to calculate them another way.

SOLVING SIMULTANEOUS – FINDING TWO PAIRS OF CO-ORDINATES

If you recall from last time, the ideal two pairs of co-ordinates to find are the following…

(𝑥 ,0)

(0 , 𝑦)

This way, we can remove one unknown from each equation and calculate the other.

We need to do this for both lines.

SOLVING SIMULTANEOUS – FINDING TWO PAIRS OF CO-ORDINATES

Let’s use the same example as before.

Graphically, solve the simultaneous equations and .

So we need two pairs of co-ordinates for each line.𝑥+𝑦=8 𝑥− 𝑦=2

…𝑦=8∴(0 ,8)

…𝑥=8∴(8 ,0)

…𝑦=−2∴(0 ,−2)

…𝑥=2∴(2 ,0)

SOLVING SIMULTANEOUS – FINDING TWO PAIRS OF CO-ORDINATES

So for we get (0, 8) and (8, 0).

For we get (0, -2) and (2, 0).Let’s connect points (0, 8) and (8, 0) first and label.(0 ,8)

(8 ,0)

𝑥+𝑦=8Next, let’s connect points (0, -2) and (2, 0) and label.

(0 ,−2)(2 ,0)

𝑥− 𝑦=2 Once again, we can see that the lines cross at (5, 3).

(5 ,3)

So .

ANSWERS

1.𝑥=1 , 𝑦=52. 𝑥=3 , 𝑦=43. 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠4.𝑥=−0.5 , 𝑦=2.755.𝑥=−6 , 𝑦=36.𝑛𝑜𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠