In mathematics, it is common to come across expressions and equations. The difference between the two is that in an expression, there is no ‘equal to’ sign whereas in an equation, there is a left-hand side and a right-hand side with ‘equal sign’ separating them. In this article, we will see what these linear equations are and how they are useful, also, how to solve such equations.
Some examples of equations are 4x = 24, x – 3 = 6. As mentioned before, equations use the equality sign, which is missing in the expressions. Both the above examples of equations are linear as the highest power of the variable x in the equation is one. Also, both these equations have one variable, ‘x.’ Such equations are called linear equations in one variable. Some examples of equations that are not linear and one variable are x2 + 3 = 5x. This is not linear because the highest power of the variable x is greater than one, but it is one variable equation. An equation like 3x + 5y = 10 is a linear equation as the highest power of the variable is one, but it is not a single variable as there are two visible variables, x, and the other one y, in the equation.
As per the definition, an equation is an equality and involves variables. On the left of the equality sign is the Left Hand Side (LHS), and on the right of the equality is the right-hand side, RHS, and as clear from the equality sign, the values of RHS and LHS are equal in an equation, but this happens for only certain values of the variables involved in the equation.
Let us have a look at the techniques used to solve a linear equation.
Example 1: Find the answer to the equation 2x – 3 = 3
Step 1- Add 3 to both sides. 2x – 3 + 3 = 3 + 3 (balance is not disturbed)
Step 2- Divide both sides by 2 so, 2 x/2 = 6/ 2 or x = 3 (required solution)
Example 2: Solve 3y + 4 = 10
Transposing 4 to RHS 3y = 10 – 4 or 3y = 6
Dividing both sides by 3, y = 2 (solution)
Example: Assume the perimeter of a given rectangle is 10 cm, and its width is equal to 4 cm. Find its length.
Solution: Let us assume the length of the rectangle to be x cm.
The perimeter of the rectangle is equal to 2 × (length + width) = 2 × (x + 4 ) = 2x + 8
The perimeter is given to be 10 cm. Therefore, 2x + 8 = 10 or 2x = 2
(dividing both sides by 2) or x = 1
The length of the rectangle is 1 cm.
Example: The present age of Ram’s father is four times the present age of Ram. After a span of 5 years, their ages will add to 70 years. Find their present ages.
Solution: Let Ram’s present age be x, as the father is four times the age of Ram, his age would be 4x. After five years, Ram’s age would be x+5, and his father’s age would be 4x + 5. Addition of two that is x + 4x + 10 = 70
5x + 10 = 70
5x = 60
X = 12
So, Ram’s age is 12 years, and his father is four times his age, so his age would be 48.
Math worksheets will be an excellent source for the students to solve more such problems and gain expertise in solving linear equations. Students can visit Cuemath website to download such worksheets for free, also these are printable and will be a fun and interactive way to practice math.