Posted In: KS3 , Year 11 maths
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In maths, an equation is a statement that two things are equal. In an equation, there are two expressions, one on each side of the equal to sign.
The followings are examples of equations:
8 + 2 = 10
7 * 5 = 35
12 - 5 = 7
15 / 3 = 5

Variables in an equation

Variables are the unknown in an equation. An equation could contain one or more unknowns. In the equation 8 + x = 10, x is the unknown variable. We can now solve this equation to determine the value of x as shown below:
8 + x = 10
x = 10 - 8
Therefore, x = 2
This process of finding the value of the unknown is called solving the equation.

Practice questions
Question: Find the value of x in the following equations.
1. 12 - x = 7
2. 13 * x = 26
3. 5 + x = 8
( Answer: 5, 2, 3 )

Simultaneous equations
Simultaneous equations are two equations with two unknowns. They are called simultaneous equations because they must be solved together in order to determine the unknowns.
For example:
Given two simultaneous equations x + y = 10 and x - y = 6, find the values of x and y.
Equation 1: x + y = 10
Equation 2: x - y = 6
Add the two equations to eliminate the unknown y
2x = 16
x= 16/2 = 8
Now replace the value of x in equation 1
8 + y = 10
y= 10 - 8 = 2
Answer: x= 8, y= 2

Solving simultaneous equations using substitution

Sometimes you can use substitution to solve two simultaneous equations to determine the value of unknowns.
For example:
Given two simultaneous equations x + y = 12 and x = 2y, find the values of x and y.
Equation 1: x + y = 12
Equation 2: x = 2y 
Replace the value of x in equation 1
2y + y = 12
3y = 12
y = 4
Now replace the value of y in equation x
x + 4 = 12
x = 8
Answer: x= 8, y= 4

Practice questions

Question: Find the value of x and y in the following equations: 3x + y = 7 and 3x - y = 5
( Answer:     Equation 1:     3x + y = 7
                    Equation 2:     3x - y = 5
    Now add the two equations
                6x = 12
                x = 2
    Replace the value of x in equation 1
                3 * 2 + y = 7
                6 + y = 7
                y= 1
)

Question: Find the value of x and y in the following equations: y - 2x = 1 and 2y - 3x = 5 
( Answer:     Equation 1: y - 2x = 1
     Equation 2: 2y - 3x = 5
    Rearrange equation 1: y = 1 + 2x
    Substitute in equation 2: 2 (1 + 2x) - 3x = 5
                    2 + 4x - 3x = 5
                    2 + x = 5
                    x = 3
    Use value of x in equation 1
                 y= 2 * 3 + 1 = 7
)

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