Simultaneous Equations

Simultaneous equations means you must solve at least two equations at the same time

Normally you will have to solve just 2 equations

3x – 2y = 16

5x + 4y = 12

REMEMBER

You are trying to find the value of x and the value of y that makes BOTH equations true

Aims for this topic:

You will know what simultaneous equations are

You will be able to solve simultaneous equations using graphs

You will be able to solve simultaneous equations using an algebraic method

You will be Learn how to solve real problems by forming and solving simultaneous equations using an algebraic method

On completion of this unit a learner should:

Know the basic underpinning mathematical techniques- methods used to manipulate and/or solve formulae, equations and algebraic expressions

PLAY IT!

There are FIVE stages to solving simultaneous equations

1. EQUATE

2. ELIMINATE

3. SOLVE

4. SUBSTITUTE

5. CHECK

5

5 5 5 5 5 55 33333 55555555555

3333333333333

33

-

-22

-22

-22

-22

-22

-22

44 4

4444

444

44

y =

-22

-22

STAGE 1: EQUATE x

3x – 2y = 16 ……….. (1)

5x + 4y = 12 ………….. (2)

Step 1: Multiply equation 1 by the Coefficient of x in equation 2 15x – 10y = 80 ……….. (3)

15x + 12y = 36 ………….. (4)

Step 2: Multiply equation 2 by the Coefficient of x in equation 1

STAGE 2: Eliminate

15x – 10y = 80 ……….. (3) 15x + 12y = 36 ……….. (4)

PLAY IT!

– 22y = 44 ……….. (5)

We must eliminate the terms containing x by adding or subtracting equation 3 and 4

STAGE 3: Solve

We have to solve equation 5 to find the value of y

-22y = 44

We do this as follows

Therefore y = -2

REMEMBERy = -2 for both equations

-2 -2 -2-2-2-2-2-2

-2-2

-2-2-2-2

USING GRAPHS

STAGE 4: Substitute

We know that y = -2 (for both equations)

Equation 1 is3x -2y = 16

PLAY IT! So to substitute We re-write equations 1 as

3 (x) -2 (-2) = 16

This gives the equation REMEMBER 3x + 4 = 16 Which gives x = 4 x = 4 for both equations

STAGE 5: Check

We know that x = 4 and y = -2

We can check this result by substituting these values in equation 2

Equation 2 5x + 4y = 12

Substituting gives5(4) + 4(-2) = 12

5 x 4 = 20 4 x –2 = -8

So 20 – 8 does equal 12

Answer is x = 4, y = -2

There are just three easy steps: Do a table of values for each equation Draw the two graphs Write down the x and y values where the graphs cross

Solving simultaneous equations using graphs

Draw the two graphs:

0

1

2

3

4

5

6

7

-3 -2 -1 0 1 2 3

x

y

y = 3 - x x + 2y = 4

Here is an example:Solve simultaneously

y = 3 - xx + 2y = 4

y = 3 -x

X -3 -2

-1

0

1 2

3

y 6 5 4 3

2 1

0

x + 2y = 4

X -3 -2

-1 0 1 2 3

y 3.5 3 2.5 2 1.5 1 0.5

Write down the x and y values where the

graphs cross :

0

1

2

3

4

5

6

7

-3 -2 -1 0 1 2 3

x

y

y = 3 - x x + 2y = 4