As we know an equation is a mathematical statement where two expressions are equated.
Examples:
2x + 5 = 11
3x - 6 = x + 12
By solution of the above equations we mean finding the values of the variable that satisfies the equation.
In this section we will take a look at methods to solve simple equations.
Operations on Equations
There are four basic rules involved in solving simple equations.
- An equal quantity can be added to both sides of the equation.
- An equal quantity can be subtracted from both the sides of the equation.
- An equal quantity can be multiplied to both sides of the equation.
- An equal non-zero quantity can divide both sides of an equation.
We may apply one or more of these rules to solve the equations.
Procedure of Solving Equation
Let’s take a look at the procedure to find solution of simple equations by considering few examples.
Example 1
Solve x + 5 = 11
We need to find the value of the variable x.
So the step here is to isolate x.
Using rule 2 subtract 5 from both the sides.
We get: x + 5 – 5 = 11 – 5
On simplification this gives: x + 0 = 6
Answer: x = 6 is the solution of the equation
Example 2
Solve 2x - 5 = 17
Step 1. Use Rule 1 and add 5 on both sides
2x – 5 + 5 = 17 + 5
The basic idea here is to bring the constants on one side of the equation.
We get: 2x + 0 = 22
i.e. 2x = 22
Step 2. Use Rule 4 and divide both sides by 2
$$ \frac2x2= \frac222 $$
The idea here is to isolate x.
We get x = 11
Hence x = 11 is the solution of the equation.
Example 3
Solve 2p + 11 – 5 = p + 14 + 5p
Step 1. Collect all the like terms on each side.
2p + 11 – 5 = p + 14 + 5p
2p + 6 = 6p + 14
Step 2. Bring constants to one side
Subtracting 6 from both sides we get:
2p + 6 – 6 = 6p + 14 – 6
2p = 6p + 8
Step 3. Bring variables on one side
Subtracting 6p from both sides we get:
2p – 6p = 6p + 8 – 6p
-4p = 8
Step 3. Isolate the variable
Divide both sides by -4
$$ \frac-4p-4= \frac8-4 $$
We get p = -2
Hence p = -2 is the solution of the equation.
Checking the Solution of an Equation
It is always a good practice to check the solution obtained.
In example 1 above we got x = 6.
To Check : Plug in x = 6 in the left side of the equation.
x + 5 = 6 + 5 = 11
we get 11 after plugging in the value x=6 which is actually the right hand side of the original question.
Therefore, x = 6 is the correct answer to the given equation.
Next, Consider example 2.
We got x = 11
Plug - in x = 11 in the left side of the equation we get:
2x - 5 = 2(11) - 5 = 22 - 5 = 17 which is also the right hand side of the original equation.
Hence our solution to the given equation is correct.
Similarly, we can check that p = -2 is also the correct solution to example 3 above.
Summary
We summarize the above procedures as follows:
1. Using the rules always bring the variables to one side of the equal sign and constants to the other.
2. Isolate the variable to get the solution of the equation.
3. Check the solution by putting the value back in the original equation and verifying
right hand side = left hand side.