**SIMULTANEOUS EQUATIONS**

**CONTENT**

- Solving Simultaneous Equations Using Elimination and Substitution Method
- Solving Equations Involving Fractions.
- Word problems.

**SIMULTANEOUS LINEAR EQUATIONS**

**Methods of solving Simultaneous equation**

- Elimination method
- Substitution method

iii. Graphical method

** **

**ELIMINATION METHOD **

One of the unknowns with the same coefficient in the two equations is eliminated by subtracting or adding the two equations. Then the answer of the first unknown is substituted into either of the equations to get the second unknown.

** **

**Example**

Solve for x and y in the equations 2x + 5y = 1 and 3x – 2y = 30

__Solution__

To eliminate x multiply equation 1 by 3 and equation 2 by 2

2x + 5y = 1 ………. eqn 1 (x (3)

3x – 2y = 30 ………… eqn 2 (x (2)

Resulting into,

6x + 15 y = 3 ………. eqn 3

6x – 4y = 60 ……….. eqn 4

Subtract eqn 3 from eqn 4

6x – 6x + 15y – (- 4y) = 3 – 60

__19y__ = __-57 __ 3

19 19

**y = -3**

** **Substitute y = – 3 into eqn 1

2x + 5 (-3) = 1

2x = 1 + 15

2x =__16__

2 2

x = 8

\ y = -3 and x = 8

** **

**Evaluation**

Using elimination method to solve the simultaneous equations.

- 5x – 4y = 38 and x + 3y = 22
- 2c-3d= -4 and 4c-3d= -14

**SUBSTITUTION METHOD**

One of the unknowns (preferably the one having 1 has its coefficient) is made the subject of the formula in one of the equations and substituted into the other equation to obtain the value of the first unknown which is then substituted into either of the equations to get the second unknown.

** **

**Example: **Solve the simultaneous equation 2x + 5y = 1 and 3x – 2y = 30

**Solution**

2x + 5y = 1……………. eq 1

3x – 2y = 30 ………….. eq 2

Make x the subject in eqn 1

__2x__ = __1 – 5y__

2 2

x = __1 – 5y__ ………… eqn 3

2

Substitute eq3 into eqn 2

__3 (1-5y)__ – 2y = 30

2

Multiple through by 2 or find the LCM and cross multiply.

__3 – 15y – 4y__ = 30

2

3 – 15y – 4y = 60

3 – 19y = 60

-19y = 60 – 3

__-19y__ =__ 57 __ 3

-19 -19

y = – 3

Substitute y = -3 into eq 3

x =__1 – 5y__

2

x = __1 – 5 (-3)__= __1 + 15__ = __16__

2 2 2

x = 8

\ x = 8, y = -3

**Evaluation**

Solve for x and y in the equations

- x + 2y = 10 and 4x + 3y = 20
- 4x-y=8 and 5x+y=19

**SIMULTANEOUS EQUATIONS INVOLVING FRACTIONS**

**Example**

- Solve the following equations simultaneously

__2__ – __1 __= 3 and __4 __ + __ 3 __ = 16

x y x y

**Solution**

__2 __ – __1__ = 3

x y

__4__ + __3__ = 16

x y

Instead of using x and y as the unknown, let the unknown be (^{1}/_{x}) an (^{1}/_{y}).

2(^{1}/_{x}) – (^{1}/_{y}) = 3 ……………. eqn 1

4 (^{1}/_{x}) – 3 (^{1}/_{y}) = 16 …………… eqn 2

Using elimination method, multiply equation 1 by 2 to eliminate x.

4(^{1}/_{x}) – 2(^{1}/_{y}) = 6 …………….. eqn 3

4 (^{1}/_{x}) + 3(^{1}/_{y}) = 16 ……………. eqn 4

__-5 ( ^{1}/_{y})__ =

__-10__

-5 -5

__1 __ = 2

y

**\**** y = ½ **

Substitute (1/y) = 2 into eqn 1

2 (^{1}/_{x}) – (^{1}/_{y}) = 3

2 (^{1}/_{x}) – (2) = 3

2(^{1}/_{x}) = 3 + 2

2 (^{1}/_{x}) = 5

__1__ = __5__

x 2

\ x = ^{2}/_{5}

\ y = ½, x = ^{2}/_{5}

**Evaluation**

- Solve for x and y simultaneously, II. Solve the pair of equations for x and y

__x __ +__y __ = 1 respectively.

2 2 2x^{-1} – 3y^{-1} = 4

__x __ – __y__ = 1½ 4x^{-1} + y^{-1} = 1

2 6

**FURTHER EXAMPLES **

Solve for x and y simultaneously: 2x – 3y + 2 = x + 2y – 5 = 3x + y.

**Solutions**

2x – 3y + 2 = x + 2y – 5 = 3x + y

Form two equations out of the question

2x – 3y + 2 = 3x + y

x + 2y – 5 = 3x + y

OR

2x – 3y + 2 = x + 2y – 5 ————- eq 1

x + 2y – 5 = 3x + y ————– eq 2

Rearrange the equations to put the unknown on one side and the constant at the other side.

2x – 3y – x – 2y = – 5 – 2

2x – x – 3y – 2y = -7

x – 5y = -7 —————- eq 3

From eqn 2

x – 3x + 2y – y – 5

– 2x + y = 5 ————- eq 4

Using substitution method solve eq 3 & 4

x – 5y = -7 —————- eq 3

-2x + y = 5 ————— eq 4

Make y the subject in eq 4.

y = 5 + 2x ————— eq 5

Substitute eqn 5 into eqn 3.

x – 5 (5 + 2x) = -7

x – 25 – 10x = -7

-9x – 25 = -7

-9x = -7 + 25

-9x = 18

x = __18__

-9

X = -2

Substitute x = – 2 into eqn 5

y = 5 + 2x

y = 5 + 2(-2)

y = 5 – 4

y = 1

\ x = -2, y = 1

**Example**

Solve the equations

5^{x – y/2} = 1 __81 ^{x}__ = 27

^{3x -y}

9

**Solution**

5^{x – y/2} = 1 ———– eq 1

__81 ^{x}__= 27

^{3x -y}———- eq 2

9

From eq 1 (using the law of indices)

5^{x – y/2} = 5^{0}

x – y/2 = 0

2x – y = 0 ———— eq 3

From eq 2.

__81 ^{x}__= 27

^{3x -y}

9

__3 ^{4x}__ = 3

^{3(3x-y)}

3 ^{2 }

3 ^{4x-2} = 3 ^{3(3x-y)}

By comparison

4x – 2 = 9x – 3y

4x – 9x + 3y = 2

– 5x + 3y =2 ——— eq 4

Solve equation 3 and 4 simultaneously

2x – y = 0 ——— eq 3

-5x + 3y =2 ———- eq 4

Using elimination method: multiply equation 3 by 3

6x – 3y = 0 ——– eq 3

-5x + 3y = 2 ———- eq 4

eq 3 + eq 4

x = 2

Substitute x = 2 into eq 3

2x – y = 0

2 (2) – y = 0

4 – y = 0

4 = 0+y

4 = y

**\**** x = 2, y=4**

** **

**WORD PROBLEMS LEADING TO SIMULTANEOUS EQUATIONS**

**Examples**

1.Seven cups and eight plates cost N1750, eight cups and seven plates cost N1700. Calculate the cost of a cup and a plate

**solution**

Let a cup be x and plate be y

7x + 8y = 1750 ————– eq 1

8x + 7y = 1700 ————– eq 2

Multiply eq 1 by 8 and eq 2 by 7 to eliminate x (cups).

56x + 64y = 14000 ———- eq 3

56x + 49y = 11900 ———- eq 4

Subtracting eq 4 from eq 3

15y = 2100

y = __2100__

15

Y = 140

Substitute y = 140 into eq 2

8x + 7y = 1700

8x + 7 (140) = 1700

8x + 980 = 1700

8x = 1700 – 980

8x = 720

x = __720__

8

x = 90

\ Each cup cost N90 and each plate cost N140

** **

- Find a two digit number such that two times the tens digit is three less than thrice the unit digit and 4 times the number is 99 greater than the number obtained by reversing the digit.

**Solution**

Let the two digit number be ab, where a is the tens digit and b is the unit digit

From the first statement,

2a + 3 = 3b

2a – 3b = -3 ………….eq1

From the second statement,

4(10a + b) – 99 = 10b + a

40a + 4b – 99 = 10b + a

40a – a + 4b – 10b = 99

39a – 6b = 99

Dividing through by 3

13a – 2b = 33 ………….eq2

Solving both equations simultaneously,

a = 3 , b = 3

Hence, the two digit number is 33

**EVALUATION**

1.The sum of two numbers is 110 and their difference is 20. Find the two numbers.

2.A pen a ruler cost #30.If the pen costs #8 more than the ruler, how much does each item cost ?

** **

**GENERAL EVALUATION AND REVISION QUESTION**

- Solve the following simultaneous equation: 3(2x – y) = x + y + 5 & 5(3x – 2y) = 2 (x –y) + 1
- Five years ago, a father was 3 times as old as his son. Now, their combined ages amount to 110years. How old are they?
- A doctor and three nurses in a hospital together earn #255 000 per month, while three doctors and eight nurses together earn #720 000 per month. Calculate (a) how much a doctor earns per month. (b) How much a nurse earns per month.
- Solve simultaneously, 2
^{x + 2y }= 1; 3^{2x+y }= 27 - Solve: 2x – 2y + 5 = 3x – 4y + 2 = -1

**WEEKEND ASSIGNMENT**

- If (x-y) log
_{10}6 = log_{10}216 and 2^{x+y}=32 , calculate the values of x and y - x=1 , y=4 b. x= 4 , y =1 c. x=-4 , y= 1 d. x=4, y= -1
- The point of intersection of the lines 3x- 2y =-12 and x + 2y = 4 is …
- (5, 0) b. (3, 4) c. (-2, 5) d. (-2, 3)
- Find the value of (x – y), if 2x + 2y =16 and 8x – 2y = 44 a. 2 b. 4 c. 5 d. 6
- If 5
^{(p +2q)}=5 and 4^{(p+3q)}=16, the value of 3^{(p+q)}is ….. a.0 b. -1 c.2 d. 1 - Given
__4x – 3y__=__11__evaluate__y__^{2}– 3x

7x – 4y 23 3 a. -2 b. 3 c. -3 d. 2

** **

**THEORY**

- Given that 2
^{1- x/y}= 1/32, find x in terms of y, and hence solve the simultaneous equations

2x + 3y – 30 = 0 and 2^{1- x/y} = 1/32 (WAEC)

- A number is made up of two digits. The sum of the digits is 11. If the digits are interchanged, the original number is increased by 9. Find the original number. (WAEC).

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