simultaneous equations problem of the day! n there are 100 animals in a zoo, some which have 2 legs...

SIMULTANEOUS EQUATIONS

Problem of the Day!

There are 100 animals in a zoo, some which have 2 legs and some have 4 legs. If there are 262 animal legs altogether, how many 4 legged animals are there?

Answer:31 four legged and69 2 legged!

Please note: This Powerpoint has been uploaded to

the wiki: http://mazenod-10max.wikispaces.com

Solving Simultaneous Equations

Q. What does the word “simultaneous” mean?

A. “at the same time”

To solve simultaneous equations you need to have the same number of equations as you have variables (unknowns).

Here we will consider cases where we have two equations and two unknowns eg x + y = 3 and 2x – y = 5

Substitution MethodThis method is generally used in situations where one of the variables (say y) is the subject or can easily be made the subject of one (or both) of the equations.

Eg 1: y = 3x + 1 and y = 5 – 4x (y is the subject in both)

Eg 2: y = 2x - 1 and 2x + 3y = 5 (y is the subject of equation 1)

Eg 3: y + 2x = 1 and 2x + 3y = 5 (y can easily be made the subject in equation 1)

Eg 4: 3y = 2x - 1 and x + 3y = 5 (x can easily be made the subject of equation 2)

Substitution MethodIn this method we will make use of the mathematical truth:

“If things equal the same thing then they must equal each other”

ie If a = b and a = c then b = c

Example:

if y = x+1 and y = p

then x+1 = p

ie we can swap (substitute) x+1 for y in equation 2 or…

Substitution Method

Example

3 yx115 yx

Label the equations (1) and (2)

(1)

(2)

Step 1

Step 2 Make one of the variables (we will use y) the subject of one of the equations (we will use equation 1)

From (1) xy 3 (1’ )

Step 3

Step 4

Substitute 3 – x for y

in equation (2)

xy 3

115 yx (2)

11)3(5 xx

Solve the equation

11515 xx11415 x

44 x1xStep 5

Substitute this value back into (1’ ) 2

13

y

(1’ )

Therefore the solution is (1, 2)

What does the solution (1,2) mean? It means that the point (1,2) is the

only solution that satisfies both equations.

x + y = 3 1 + 2 = 3 x + 5y = 11 1 + 5(2) = 11 Can you interpret this graphically?

Another example

934 yx


(1)

(2)

Step 1

Step 2 Equation (2) already has x the subject

12 yx

Step 3

Step 4

Substitute 2y + 1 for x in equation 1

(2)

93)12(4 yy

Solve the equation

9348 yy945 y55 y1yStep 5

Substitute this value back into (2 ) 3

1)1(2

x


934 yx (1)

12 yx

A more difficult example

2321245 yxyx and

Solve the simultaneous equations:

To get started: Which variable in which equation is the easiest to make the subject?

I will choose to make x the subject in equation 2

)2.......(232 and)1.....(1245 yxyx

Substitute this for x in equation 1 124

2

325

yy

10247 y2y

Therefore the solution is (–4, –2)

2

32 yx

2481510 yy

Substitute y = -2 into (2’) 2

)2(32 x

4x

Making x the subject in equation 2 gives:

Solve for y

Elimination Method

This method is best used when you have equations in the form:

3 yx115 yx


(1)

(2)

Step 1

Elimination Method

Best used when you have equations in the form:

3 yx115 yx


(1)

(2)

Step 1

3 yx115 yx

Which variable has the same coefficient?

x

As the sign of x is the same, we subtract equation (2) from equation (1)

Step 2

(1)

(2)Step 3

840 y84 y

2yStep 4x is eliminated, solve for y

Step 5Substitute the value for y into equation (1) and solve for x

32 x1x


What does the solution mean ?

We had two equations and two unknowns, x and y

3 yx 115 yxand

The solution (1, 2) gives the values of x and y that will make both sentences true.

Another example

832 yx143 yx


(1)

(2)

Step 1

Which variable has the same coefficient?

y

Step 2

As the sign of the 3y is different, we add equation (2) to equation (1)

(1)

(2)Step 3

603 x63 x2x

Step 4y is eliminated, solve for x

Step 5Substitute the value for x into equation (1) and solve for y

83)2(2 y123 y

Therefore the solution is (-2, 4)

832 yx143 yx

4y

A more difficult example

1032 yx265 yx


(1)

(2)

Step 1

Which variable has the same coefficient? Neither

Step 2

We can make x have the same coefficient if we multiply equation (2) by 2

As the sign is the same, we subtract equation (2’) from equation (1)

(1)

(2’)Step 3

427 y6yStep 4

x is eliminated, solve for y

Step 5Substitute the value for y into equation (2) and solve for x

26)6(5 x2630 x

Therefore the solution is (-4, 6)

4x

1032 yx52102 yx

Try questions 1 to 4 on the worksheet

A more difficult example 2321245 yxyx and

Using elimination

Multiply (1) by 3 and (2) by 4

361215 yx (1’)

1245 yx (1)(2)232 yx

8128 yx (2’)

Subtract (2’) from (1’)

2807 x4x

Substitute into (1)

124)4(5 y20124 y

2y

Therefore the solution is (-4, -2)

2321245 yxyx andUsing substitution

From (2)

1245 yx (1)(2)232 yx

Substitute (2’) into (1)

1242

325

yy

10247 y2y


2

32 yx

2481510 yy

Substitute y = -2 into (2’)

2

)2(32 x

4x

(2’)

A more difficult example 2321245 yxyx and

Using elimination

Multiply (1) by 3 and (2) by 4

361215 yx (1’)

1245 yx (1)(2)232 yx

8128 yx (2’)

Subtract (2’) from (1’)

2807 x4x

Substitute into (1)

124)4(5 y20124 y

2y

Therefore the solution is (-4, -2)

2321245 yxyx andUsing substitution

From (2)

1245 yx (1)(2)232 yx

Substitute (2’) into (1)

1242

325

yy

10247 y2y


2

32 yx

2481510 yy

Substitute y = -2 into (2’)

2

)2(32 x

4x

(2’)

Step 3

Step 4

Substitute the new equation into the other equation in this case equation (2)

xy 3

115 yx (2)

11)3(5 xx

Solve the equation

11515 xx11415 x

44 x1xStep 5

Substitute this value back into (1’ ) 2

13

y

(1’ )


Step 3

Step 4

Substitute equation (2) into equation (1)

(2)

93)12(4 yy

Solve the equation

9348 yy945 y55 y1yStep 5

Substitute this value back into (2 ) 3

1)1(2

x


934 yx (1)

1 2 y x

Complete the questions on the worksheet

We wish to thank our supporters:

simultaneous equations problem of the day! n there are 100 animals in a zoo, some which have 2 legs...

Documents