Simultaneous Equations - problem solving

Equations that must be solved at the same time are simultaneous equations. In this tutorial, I am going to explain how a pair of simultaneous equations are formed from a given problem and then solve it. If you go through the worked examples in the order given, you will have enough experience to deal with any question involving simultaneous equations.

I assume you know how to solve simultaneous equations, either by elimination method or substitution method. If you want to brush up on your skill, please go through this tutorial first, and then use this one. If you think you need a bit more practise use the following question generator for the task.

 

Algebra Equation Generator

You can generate as many questions as you want with the following programme, along with the answers. First generate the question, then work them out and check with the answer.

 

E.g.1

The sum of two numbers is 6 and the difference is 2. Find the numbers.
Let the numbers be x and y.
x + y = 6 1
x - y = 2 2
1 + 2 => 2x = 8
x = 4
Sub in 1=> 4 + y = 6
y = 2
The numbers are x = 4 and y = 2.

E.g.2

The sum of two books and a pencil is £6.00. The difference of cost between 3 books and 2 pencils is £2.00. Find the cost of a book and a pencil.
Let the cost of a book be x and that of a pencil be y.
2x + y = 6 1
3x - 2y = 2 2
1 X 2 => 4x + 2y = 123
2 + 3 => 7x = 14
x = 2
Sub in 1
4 + y = 6
y = 2
The cost of a book and a pencil is £2.00 each.

E.g.3

If I double a number and add three times a second number, the answer is 1. If I multiply the first number by 3 and take away twice the second number, the answer is 8. Find the numbers.
Let the numbers be x and y.
2x + 3y = 1 1
3x - 2y = 8 2
From 1 - 3y => 2x = (1 - 3y)
x = (1 - 3y)/2
Sub in 2 => 3(1 - 3y)/2 - 2y = 8
X 2 => 3(1 - 3y) - 4y = 16
3 - 9y - 4y = 16
3 - 13y = 16
-3 => -13y = 13
y = -1
Sub in 1 => 2x - 3 = 1
+ 3 => 2x = 4
x = 2
The numbers are x = 2 and y = -1.

E.g.4

The sum of twice the cost of a box biscuits and the cost of a box chocolates is £8.00. The difference between the cost of box of chocolates and the box of biscuits is £1.00. Find the cost of each.
Let the cost of the box chocolates and the box of biscuits be y and x respectively.
2x + y = 8 1
y -x = 1 2
1 => y = 8 - 2x
Sub in 2 => y = 8 - 2x - x = 1
8 - 3x = 1
-8 => -3x = -9
:- -3 => x = 3
Sub in 1 => 6 + y = 8
-6 => y = 2
The cost of box of chocolates =£2.00 and that of biscuits = £3.00.

Simultaneous Quadratic Equations

These equations are simultaneous as there are two unknowns in them; since one of the unknown is in quadratic form, they are quadratic too. Therefore, these equations have two sets of solutions, one for each unknown.

E.g.1

The equation of a circle is x² + y² = 45. It intersects with, y = 2x, at two points. Find the coordinates of the points of intersection.
x² + y² = 45 1
y = 2x 2
Sub y in 1 => x² + 4x² = 45
5x² = 45
:- 5 => x² = 9
x = ± 3
Sub in 2 => y = ±6
Solutions: (3,6); (-3,-6)

Solving Simultaneous Equations - fully interactive

The following applet help you solve simultaneous equations instantly. Just type in the two equations into the text boxes, exactly in the form shown, and then press enter. You may move the grid to see the point of intersection of the two lines.

 

Questions for Practice

Now, in order to complement what you have just learned, work out the following questions:

1. The sum of two numbers is 18. The difference is 4. Find the numbers.
2. A straight line passes through (3 , -4) and (5 , 8). Find the equation of the line.
3. The cost of 3 DVD's and 4 CD's is £62.00. The cost of 4 DVD's and 3 CD's is £64.00. Find the cost of each.
4. A diver swims downstream a distance of 40 miles in two hours. If he swims upstream, he can only move 16 miles during the same time. Find his swimming speed and the speed of the river.
5. Half the difference between two numbers is 8. The average of the numbers is 12. Find the numbers.
6. Nicole has 23 notes of £20 and £5 in her hand bag. The amount of money she has in the bag is £340.00. Find the number of notes of each type.
7. There are two angles on a straight line. One angle is 15 more than twice the other. Find the size of each angle.
8. The sum of ages of an uncle and his nephew two years ago was 40. In two years time from now, the age of the uncle will be three times that of his nephew by then. Find their ages in 7 years time.
9. The curve, px² + qx, passes through (3,6) and (1,-2). Find the values of p and q.
10. The numerator of a fraction is 3 smaller than its denominator. If both the numerator and denominator are increased by by 1, the fraction is 5/8. Find the original fraction.
11. A car covers a distance of 220 miles at 3o mph and 20 mph respectively. Find the time taken for each part of the journey.
12. The curve, px² + qx + r, passes through (0,5) and (2,5) and (3,11). Find the values of p, q and r.

 

Answers

Move the mouse over, just below this, to see the answers:

1. 11,7
2. y = 6x - 22
3. 10,8
4. 14,6
5. 20,4
6. 15,8
7. 125, 55
8. 34, 10
9. 2, -4
10. 4/7
11. 6, 2
12. 2, -4, 5