1. What are Simultaneous Equations?
Simultaneous equations are a pair of equations which consists of more than one variable or unknown value (usually two).
Examples of simultaneous equations are 2x + y = 9 and x + 2y = 9.
To solve simultaneous equations, we need to find the value of the variables or unknowns. In the above example, we would solve for x and y. They are called simultaneous equations because the equations are solved at the same time.
1.1 How to Solve Simultaneous Equations?
Simultaneous equations are solved using a variety of ways, but for the 11+ exams, we prefer to solve them using the method of elimination.
Let us try to solve 2x + y = 9 and x + 2y = 9.
First step is to number the equations. Here, 2x + y = 9 …(1) and x + 2y = 9 …(2)
Now check the numbers in front of the variables. x has 2 and 1 while y has 1 and 2. To eliminate the variable, we require one of the variables to have the same numbers in front. Since both of the variables have different numbers, we need to multiply one of the equations.
Multiply (1) with 2, to get, 4x + 2y = 18 …(3)
Now we can subtract (2) from (3),
(Note: If the signs for y were different, we would add the equations)
We can now solve the equation 3x = 9 to get, x = 3. The next step is to substitute the value of x in either (1) or (2). Let us substitute it in (1) and solve for y.
23 + y = 9
6 + y = 9
y = 3
The solution for the equation is x = 3, and y = 3.
1.2 Example Questions on Simultaneous Equations
Example: Solve the following simultaneous equations
4x + 2y = 10
3x + 3y = 9
Solution : 4x + 2y = 10 …(1)
3x + 3y = 9 …(2)
Multiply (1) by 3
12x + 6y = 30 …(3)
Multiply (2) by 2
6x + 6y = 18 …(4)
Subtracting (4) from (3),
x = 12 ÷ 6 = 2
Substituting x = 2 in (1),
4 × 2 + 2y = 10
8 + 2y = 10
2y = 2
y = 2 ÷ 2 = 1
The solution for the equation is x = 2, and y = 1.
Example 2: Solve the following simultaneous equations
a + 2b = 5
a - 2b = - 3
Solution: a + 2b = 5 …(1)
a - 2b = - 3 …(2)
Adding (1) and (2),
a = 2 ÷ 2 = 1
Substituting a = 1 in (1),
1 + 2b = 5
2b = 4
b = 2
The solution for the equation is a = 1, and b = 2.
