Simultaneous Equations - Examples and Questions with Answers

algebra Jun 03, 2021

What are Simultaneous Equations?

This article will introduce one of the most important topics in 11+ Exams (GL and CEM): Simultaneous Equations. The reader will be introduced to various methods used in solving Simultaneous Equations (Elimination Method & Substitution Method). We will also be solving some word problems based on real-life issues.

Linear Equations in two Variables

An equation of the form ax + by + c = 0 where a, b and c are real numbers and a and b are non-zero is called a general linear equation in two variables x and y.
For example, x + y -3 = 0 is a linear equation in the two variables (Unknowns) x and y

x = p and y = q is a solution of the linear equation ax + by + c = 0 if and only if ap + bq + c = 0, where p & q are real numbers.

Simultaneous Linear Equations

The term Simultaneous Equations refer to a condition where two or more variables are linked to each other via an equal number of equations.
Let us consider two linear equations in two variables,

a1x + b1y = c1

a2x + b2y = c2

These two equations are said to form a system of simultaneous Linear Equation in two variables x and y.
A solution to a system of two Simultaneous Equations in two variables is an ordered pair of numbers that satisfy both the equations.

For example:

x + y = 3

2x - 5y = -1

x = 2, y = 1 is the solution of the system of Simultaneous Equations because when we substitute these values back in the equation, the values satisfy both the equations.
A System of Simultaneous Equation can have an unlimited number of solutions. If there is only ONE solution, then the system of linear equations is said to be Consistent and Independent.

To find the solutions to the various system of Equations, one can always use the Simultaneous Equations Calculator.

Solving Simultaneous Equations 

The various methods employed in solving System of Equation in 11+ exams are:

  1. Elimination Method
  2. Substitution Method
  3. Cross Multiplication Method

We shall discuss these methods one by one.

Elimination Method to solve Simultaneous Equations

Elimination by addition:

In this method, we eliminate one variable by adding both the equations and then solve them for the other variable.

Example: The sum of the two numbers is 14 and the difference is 4. Find the numbers.

Solution: 


 

Elimination by Subtraction:

In this method, we eliminate one variable by subtracting both the equations and then solve them for the other variable.

Example: Solve 2x + y = 14, 2x - y = 4. 

Solution:


 

Practice Questions:

  1. Solve: x + 2y = 28
                x + y = 20
  2. Solve: 2x - 2y = 2
                3x + 2y = 3
  3. Solve: x/5 + 2y/5 = 12
                x/5 - 3y/5 = 10
  4. The sum of the two numbers is 26. If the difference between them is 12. Find the two numbers.
  5. If on the first week A and B together walk for 20 miles. In the second week, A walks 2 miles more than B. Find the distance walk by each of them in two weeks.

Answers:

  1.  x = 12, y = 8
  2.  x = 1, y = 0
  3. x = 56, y = 2
  4. 17, 9
  5. A: 11 miles, B: 9 miles

Elimination by Multiplying one Equation:

In this method, we eliminate one variable by making the coefficients of any one variable equal in both equations. By multiplying one equation.

Example: The sum of two numbers is 20. And the difference of twice the first number and second number is 4. Find the numbers.

Solution:


 

Practice Questions:

  1. A movie ticket cost £3 for adults and £2 for children. On the weekends 2000 people watched the movie with a collection of £ 5000.find the number of adults and children?
  2. Sam has 40 coins, consisting of 20p and 10p, which total £6.50. How many of each does she have?
  3. Two pizzas and one Burger cost £45 and 3 pizzas and four Burgers cost £ 55. Find the cost of pizza.
  4. A two-digit number is 7 times the sum of its digits. The number formed by reversing the digits is 18 less than the original number. Find the number.
  5. Twice one number minus three times a second is equal to 2, and the sum of these numbers is 11. Find the numbers.

 Answers:

  1. Children: 1000; Adults: 1000
  2. 20p: 25 coins; 10p: 10 coins
  3. Pizza: £22; Burger: £1
  4. 42
  5. 7 and 4

Substitution Method to solve Simultaneous Equations

To apply the substitution method, we can follow the steps.

  1. Solve any one of the given equations for one of the variables, whichever is convenient.
  2. Substitute that value of the variable in the other equation.
  3. Solve the resulting single variable equation. Substitute this value into either of two original equations or solve it to find the value of the second variable.

Example: The sum of two numbers is 43, if the larger is doubled and the smaller is tripled, the difference is 36. Find the two numbers.

Solution:

 


 

Practice Questions:

  1. Solve:
                 x + 3y = 9
                 3x - 3y =15
  2. The sum of thrice the first number and twice the second numbers is 59 and the difference of twice the first number and thrice the second number is 22. Find the numbers.
  3. The cost of 5packs of cookies and 7 packs of chocolates is £153. And the cost of 7 packs of cookies and 5 packs of chocolates is £147. Find the cost of each pack of cookies and chocolates.
  4. 6 years hence Jack’s age will be three times his daughter’s age, and three years ago he was nine times as old as his daughter. Find their present ages.
  5. If twice the age of George is added to the age of his father, the sum is 64. But if twice the age of the father is added to the age of George, the sum is 92. Find the ages of George and his father.

Answers:

  1. x = 6; y = 1
  2. 17 and 4
  3. Cookies: £11; Chocolate: £14
  4. Jack: 30 years; Daughter: 6 years
  5. George: 12 years; Father: 40

Cross Multiplication Method to solve Simultaneous Equations

Let us consider two linear equations in two variables:

To solve the simultaneous equation by using the cross multiplication method, Write the coefficient of the pair of the linear equation as:

The arrows between the two numbers indicate that they are to be multiplied. The down arrows ↘️ show the term with a plus sign and the up arrow ↗️ shows the term with a negative sign.
The solution is given by:

Example: At a certain time in Zoopark, the number of heads and the number of legs of Zebra and human visitors was counted, and it was found that there were 41 heads and 136 legs. Find the total number of Zebras and human visitors?

 Solution:


 

Practice Questions

  1. Solve:
              3x + 2y - 4 = 0
              8x + 5y - 9 = 0
  2. Solve:
              2x + y - 5 = 0
              3x + 2y - 8 = 0
  3. Solve:
              x + y = -1
              2x - 3y = -5
  4. Solve:
              x– 3y -7 = 0
              3x - 3y – 15 = 0
  5. A chemist has one solution which is 50% acid and a second which is 25% acid. How much of each should be mixed to get 10 litres of 40% acid solutions.

Answers:

  1. x = -2, y = 5
  2. x = 2, y = 1
  3. x = -8/5, y = 3/5
  4. x = 4; y = -1
  5. 6 liters of 50% acidic solution and 4 litres of 25% acidic solution

Tips to solve Simultaneous Equations Word Problems

  • Solving word problems involves two steps:
    1. First converting the words of the problem into equations.
    2. Second, Solving the resulting equations
  • While solving Simultaneous Equations word problems, the following general suggestions should prove helpful:
    1. Read the statement of the problem thoroughly, and determine what quantities must be found
    2. Represent the unknown quantities by alphabets.
    3. Form the equations using the given information.
    4. Solve the resulting equations using the appropriate method

 

To learn more about 11+ Advanced Algebra topics, word problems and how to solve them, view our video lessons on Advanced Algebra Word Problems for 11+ Exams.

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