Simultaneous equations

•Sketching straight lines

•Solving simultaneous equations by straight line graphs

•Solving simultaneous equations by substitution

•Solving simultaneous equations by elimination

•Knowing two points which lie on a line find the equation of the line

•Using simultaneous equations to solve problems

•Simultaneous equation by substitution

•Problems from credit past papers

Sketching straight lines

Equations of the type

y = x ; y = x + 3 ; y = 4x - 5; x + y = 2 are equations of straight lines . The general equation of a straight line is

y = ax =b

By finding the coordinates of some points which lie on the straight lines, plotting them and joining them up the graph of the straight line can be drawn.

To draw the graph of a straight line we must find the coordinates of some points which lie on the line.We do this by forming a table of values . Give the x coordinate a value and find the corresponding y coordinate for several points

Table of values

x

y=2x+1

Make a table of values for the equation y = 2x + 1

0 1 2 3

1 3 5 7

Now plot the points on a grid and join them up

So (0,1) (1,3) (2,5) and (3,7) all lie on the line with equation y=2x + 1

x

y

0 1 2 3 4

7

6

5

4

3

2

1

Plot the points (0,1) (1,3) (2,5) and (3,7) on the grid

..

..

Now join then up to give a straight line

All the points on the line satisfy the equation y = 2x + 1

Sketching lines by finding where the lines cross the x axis and the y axis A quicker method

Straight lines cross the x axis when the value of y = o

Straight lines cross the y axis when the value of x =0

Sketch the line 2x + 3y = 6

Line crosses x axis when y = 0

2x + 0 =6

2x =6

x =3

at ( 3,0)

Line crosses y axis when x = 0

0 + 3y = 6

3y =6

y = 2

at ( 0,2)

Ex 2 page 124

Plot 0,2) and (3,0) and join them up with a straight line

x

y

0 1 2 3 4

7

6

5

4

3

2

1

..

Now join then up to give a straight line

All the points on the line satisfy the equation 2x + 3y = 6

Solving simultaneous equations by straight line graphs

Two lines either meet at a point intersect or they are parallel they never meet

To find where two lines meet draw the graphs of both lines on the same grid and read off the point of intersection

Find where the lines x + y = 5 and x – y =1 intersect

x + y = 5 get two points that lie on each line or better still three

(0,5) and (5,0) and (1,4)lie on the line x + y = 5

x – y =1

(0,-1) and (1,0) and ( 4,3) lie on the line

Now plot the points and find where the two lines meet

x

y

0

5 0

51

4

x

y

0

-1 0

14

3

x

y

1 2 3 4 5

7

6

5

4

3

2

1

0

-1

.

.

.

x + y =5

..

.x + y =1

X

(3,2)

The lines intersect at (3,2)

Solving simultaneous equations algebraically

Whoopee no more drawing

graphs

But you will have to learn a strategy

to solve the equations

Yikes!

Elimination method

Solve the simultaneous equations algebraically

x + y = 8

x - y = 4

HOW DO I SOLVE TWO EQUATIONS WITH TWO

UNKNOWNS ?

ADD the two

equations together

2x = 12

ADD This gives one equation with one unknown

which is easy to solve

x = 6

How do I find y. It is gone.It has

been eliminated ?

SUBSTITUTE THE VALUE OF X INTO ONE OF YOUR

EQUATIONS

x + y = 8

6 + y = 8

y = 2

x = 6

Now I have the solution

Am I correct?

Check by putting the values of x and y into , the other equation, x-y,and see if

you get 4

6 – 2 = 4

Solution is x =6 and y = 2

Solve the simultaneous equations

x + 3y = 7

x - 3y = -5

ADD 2x = 2

x = 1

Substitute x =1 into x + 3y = 7 1 + 3y = 7

3y =6

y = 2

Remember to check

1 – 3 X 2

= 1 -6

= -5 Solution is x = 1 and y = 2

Solve the simultaneous equations

3x + y = 8

x + y = 4

If I add I get

another equation

in x and y

3x + 2=12no useMultiply

one of the equations by ( -1)

It changes the sign

of everythin

g

3x + y = 8

-x -y = -4

ADD2x = 4x = 2

SUBSTITUTE Put x=2 into x + y =4

x + y = 4 2 + y = 4

y = 2

3X2 + 2 = 8

Solution x = 2 and y = 2

check

Using simultaneous equations to solve problems

example

Slightly more difficult

But not for you IF you have learned the

work so far

Solve the simultaneous equations

3x + 2y = 29

2x + 3y = 26

Adding gives 5x + 5y = 55 no

use

Multiplying by a negative does not eliminate any of

the letters

Could I multiply and then

add?That ‘s good let’s try it but

be careful

Remember when we add numbers together which are the negatives of each other

they are eliminated

3x + 2y = 292x + 3y = 26

Multiply the equations by two suitable numbers so that the coeficients of x or y are the negatives of each other

3x + 2y = 292x + 3y = 26

X 3

X -2

Giving two new equivalent equations

9x +6y = 87

-4x + -6y = -52 Now add

Solve the following simultaneous equations algebraically

We choose y to be eliminated

9x +6y = 87

-4x + -6y = -52

ADD 5x = 35

x = 7

substitute

9 X 7 + 6y = 8763 +6y = 87

6y = 24

y = 4

check

-4 X 7 - 6X 4

= -28 –24

= -52

Solution x = 7 and y = 4 Ex 7B p 139

Remember practice makes perfect

x = 7 into 9x+6y = 87

Using simultaneous equation to solve problems

Problem Solving With Simultaneous EquationsExample : The problemFor the cinema

4 adults’ tickets and 2 children tickets cost £28

2 adults’ tickets and 5 children’s tickets cost £26

Introduce LettersLet the cost of adult ticket = £A

Let the cost of children’s ticket = £C

Write the equations

.......(1) 2652 CA .......(2) 2824 CA

Find the cost of each kind of ticket.

Solve the Problem using simultaneous equations

X2 . 2652 CA1)- ( X 2824 CA

28)2 (- 4-

. 52104

CA

CA

(1) into 3 Substitute C

Conclusion

Adult Ticket costs £5.50

Children's tickets cost £3

3

248 ADD

C

C

55 A

26352 A26 152 A

112 A

strategy1. Read the problem

2. Introduce two letters

3. Write two equations that describe the information

4. Solve the problem using simultaneous equations

5. check

Substitution methodThis an excellent method of solving simultaneous equations when the equations are given to you in a certain form

Example 1 Solve the equations y = 2x and y = x + 10

We take the equation y = x + 10And substitute 2x for y 2x = x + 10

Take out y and replace it with 2x

Now solve the equation

x = 5

Sub x=5 into y=2x

y = 10

Solution x = 5 y = 10

Example 2

Solve the equations y = 2x –8 and y – x = 1

Sub y =2x-8 in the equation

2x –8 -x = 1

x – 8 = 1

x = 9

Sub x = 9 in the equation y = 2x -8

y = 18-8

y = 10

Solution x = 9 and y =10 Check 10 –9 =1

y - x = 1

Problems with simultaneous equations from past papers

1. The tickets for a sports club cost £2 for members and £3 for non- members

a) The total ticket money collected was £580

x tickets were sold to members and y tickets were sold to non-members.

Use this information to write down an equation involving x and y

2x + 3y =580

b) 250 people bought raffle tickets for the disco. Write down another equation involving x and y.

x + y = 250

c ) How many tickets were sold to members ?

We now have two equations in x and y so let’s solve them simultaneously

2x + 3y = 580

x + y = 250

2x + 3y = 580

x + y = 250

X 1

X (-2)

2x + 3y = 580

-2x + (-2 )y = -500ADD

y = 80

Sub y = 80 into x + y = 250

x + 80 =250

x = 170

check

2 x170 + 3x80 =340 + 240 =580

170 tickets were sold to members

2. Alloys are made by mixing metals.Two different alloys are made using iron and lead. To make the first alloy, 3 cubic cms of iron and 4 cubic cms of lead are used. This alloy weighs 65 grams

a ) Let x grams be the weight of 1 cm3 f iron and y grams be the weight of 1 cm3 of lead Write down an equation in x and y which satisfies the above equation.

3x + 4 y = 65

To make the second alloy, 5 cm3 of iron and 7cm3 of lead are used . This alloy weighs 112 grams.

b ) Write down a second equation in x and y which satisfies this condition

5x + 7y = 112

C ) Find the weight of 1 cm3 of iron and the weight of 1cm3 of lead.

3x + 4 y = 65 5x + 7y = 112

X (-5)X 3

-15x + -20 y = -325

15x + 21y = 336ADD

y = 11

SUB y = 11 into 3x + 4y = 65

3x + 44 = 65

3x = 21

x = 7

Check 5x7 + 7x11

= 35 + 77= 112

1cm3 of iron weighs 7gms and 1cm3 of lead weighs 11 gms

3. A rectangular window has length , l cm and breadth, b cm .

A security grid is made to fit this window. The grid as 5 horizontal wires and 8 vertical wires

a ) The perimeter of the window is 260 cm. Use this information to write down an equation involving l and b .

2 l + 2 b = 260

b) In total, 770 cm of wire is used. Write down another equation involving l and b

5 l + 8 b = 770

2 l + 2 b = 260

5 l + 8 b = 770

Find the length and breadth of the windowX (4)

X (-1)

ADD

8 l + 8 b = 1040

-5 l + (-8) b = -770

3 l = 270 l = 90

SUB l = 90 into 2 l + 2 b = 260

180 + 2b = 2602b = 80

b = 40

5x90 +8x40 = 450 + 320=770

Length is 90cm and the breadth is 40 cm

4. A number tower is built from bricks as shown in fig 1.The number on the brick above is always equal to the sum of the two numbers below.

912 -3

8 4 -73 5 -1 -6

fig1

3416 18

5 11 7-2 7 4 -11

Find the number on the shaded brick in fig 2

Fig 2

34

-3

p q -5 2

In fig 3, two of the numbers on the bricks are represented by p and q

Show that p + 3q = 10

Fig 3

p + 2q –5 + q -8 = -3Adding the numbers in the second row to equal the top row

simplifyingP + 3q – 13 = -3

P + 3q = 10

p+q q -5 -3

p +2q -5 q-8

14

2q -2 5 -p

Use fig 4 to write down a second equation in p and q

fig 42q-2 3 5-p

2q+1 8-p

Adding the numbers in the second row to equal the top row 2q+1+8-p =14

simplifying2q +9 – p = 14

2q – p = 5

d) Find the values of p and q

3q + P = 10

2q – p = 5

3q + P = 10

2q - p = 5

ADD5q = 15

q = 3

Change the terms around q under q and p under p

SUB q = 3 into 2q –p =5

6 –p = 5

p=1

Check 3x3 +1 = 10

p = 1 and q = 3

Remember to check

5. A sequence of numbers is 1 ,5 ,12 ,22 ,………Numbers from this sequence can be illustrated in the following way using dots

First number ( N = 1)

•

Second number (N=2)

• • • • •

Third number (N=3)

• • • • •

• •

• •• • •

Fourth number (N=4)

• • • • •

• •

• •• • •

• •

• • • •• •

• •

a )Write the fifth number in the pattern

N 1 2 3 4 5

D= no of dots

1 5 12 22 35

Form a table of values

When N= 1 D = 1 When N=2 D = 5

b) The number of dots needed to illustrate the nth number in this sequence is given by the formula

D =aN² - bN

Find the values of a and b

N = 1 D = 1 When N=2 D = 5

D =aN² - bN

Form two equations by substituting the values of n and D into the equation D =aN² - bN

N = 1 D = 1 1 = a - b

N=2 D = 55 = 4a -2b

Now solve simultaneously

1 = a - b

5 = 4a -2b

X (-2)

X1

-2 = -2a + 2b

5 = 4a - 2b

ADD 3 = 2aa = 3/2

SUB 1 = 3/2 -b

b = 1/2

Check

4x3/2-2x1/2

=12/2-2/2 =10/2=5

a = 3/3 b= 1/3