Dr. J's Maths.com
Where the techniques of Maths
are explained in simple terms.

Algebra - Simultaneous equations - basic techniques.
Test Yourself 1 - Solutions.


 

Solve the following equations simultaneously by using the equation with 1 in front of the variable:

1. 2x - y = 3

9 = x + y

y = 2x - 3 (from eqn 1)

Sub into Eqn 2:

9 = x + 2x - 3

3x = 12

x = 4

y = 5

 

 2. 8 = 3x + y

y - 3 = 2x

y = 2x + 3 (from eqn 2)

Sub into Eqn 1:

8 = 3x + 2x + 3

5x = 5

x = 1

y = 5

 3. x = 11 - 3y

x + y = 7

Substitute Eqn 1 directly into Eqn 2:

11 - 3y + y = 7

2y = 4

y = 2

x = 5

 
4. y = 2x + 1

x - 2y - 4 = 0

Substitute Eqn 1 directly into Eqn 2:

x - 2(2x + 1) - 4 = 0

-3x - 2 - 4 = 0

3x = -6

x = -2

y = -3

 5. 2x - 3y = -1

x + 2y = 10

x = 10 - 2y
(from eqn 2)

Sub into Eqn 1:

2(10 - 2y) - 3y = -1

20 - 4y - 3y = -1

7y = 21

y = 3

x = 4

 6. 5x + 2y = -8

3x + y = -4

y = -3x - 4
(from eqn 2)

Sub into Eqn 1:

5x + 2(-3x -4) = -8

-x = 0

x = 0

y = -4

Equal coefficients for one variable:

7. 4a - 3b = 11

4a + 2b = 10

Subtract Eqn 2 from Eqn 1:

-5b = 1

b = -0.2

Substitute into Eqn 2:

a = 2.6

 

 8. 5e - 3f = 20

2e + 3f = 15

Add the 2 equations:

7e = 35

e = 5

Substitute into Eqn 2:

3f = 15 - 10

f = 5/3

 9. 5p - q = 24

2p - q = 3

Subtract Eqn 2 from Eqn 1:

3p = 21

p = 7

Substitute into Eqn 2:

14 - q = 3

q = 11

 
10. 11a + 9b = 34

5a - 9b = -50

Add the 2 equations:

16a = -16

a = -1

Substitute into Eqn 2:

9b = 34 - 11(-1)

b = 5

 11. 3y - 4x - 1 = 0

2y + 4x - 14 = 0

Add the 2 equations:

5y = 15

y = 3

4x = 14 - 6

x = 2

 

 12. 5p + q = 10

2p + q = 7

Subtract Eqn 2 from Eqn 1:

3p = 3

p = 1

Substitute into Eqn 2:

q = 5

 

Applied questions (mainly of the "break-even" type).

13. Two brothers Chuck and Dave drive vehicles between Sydney and Gunnedah - a distance of 430 kms. Chuck drives a truck which leaves Gunnedah at the same time as Dave leaves Sydney in his car.

Chuck drives the truck at an average speed of 60 kph while Dave drives his car at an average speed of 80 kph (its a sporty grey model).

After what time will the brothers meet up to have a coffee and a talk?

SOLNS IN TYS 2