Before going to the actual topic, i.e, linear equation in one variable, let me introduce you to the basics. Basically there are two thing in mathematics, namely, expression and another thing ‘equation’. An algebraic expression is a mathematical phrase that can contain numerics, variables and operators such as +, -, *, /. For example, 3x + 9 is a mathematical expression.
Now coming to equations, equations are similar to expression except for that equations contain ‘equal to’ operator with some other expressions. Thus, an equation is a statement of an equality containing one or more variables. Solving the equation consists of determining which values of the variables make the equality true. Variables are the unknown part of an equation or expression. For example, 4x + 15 = 20 is an equation in one variable, while 3x + 4y = 15 is an equation in two variables i.e., ‘x’ and ‘y’.
Now moving onto the actual topic, linear equation is an equation that gives a straight line when plotted on a graph. Linear equation in one variable is an equation with one unknown quantity which when plotted on the graph gives a straight line.
Definition: If an equation involves only one variable and the highest index of power of that variable is 1, the equation is called a linear equation in one variable.
Following are some examples of linear equation in one variable:
(i) 2x = 8
(ii) 4y = 9
(iii) 3z = 7
(iv) 2x + 4= 7
(v) 81x + 45 =123
All the above mentioned examples have only one variable and are linear in nature. So, they are known as linear equation in one variable.
The equation x2 = 7x + 5 is not a linear equation because the highest index of power of the variable x in it is 2.
Again, x + 5y = 10 is a linear equation in two variables x, y but not in one variable, x or y.
The general form of a linear equation in one variable x is ax + b = 0, a ≠ 0 or px = q, p ≠ 0.
Framing linear equation in one variable from given word problem:
Steps involved in framing of linear equation in one variable from the given word problem are as follows:
Step I: first of all read the given problem carefully and note down the given and required quantities separately.
Step II: Denote the unknown quantities as ‘x’, ‘y’, ‘z’, etc.
Step III: Then translate the problem into mathematical language or statement.
Step IV: Form the linear equation in one variable using the given conditions in the problem.
Sep V: Solve the equation for the unknown quantity.
Now let us try to form some linear equations from given problems.
1. The sum of two numbers is 25, one of the numbers is twice the other. Find the numbers.
Let one of the number be ‘x’.
It is given that 2nd number is two times the first number. so 2nd number = 2x.
Now sum of two numbers = 25.
Now when we convert the statement into mathematical statement, then the equation becomes, x + 2x = 25. So, 3x = 25 is our required linear equation in one variable.
2. The difference between two numbers is 70. If the numbers are in ratio 3:5. Then, find the numbers.
Let the common ratio be ‘x’.
The 1st number = 3x and 2nd number = 5x.
Now it is given that the difference between them is 70. So, converting the statement into mathematical statement we get,
5x – 3x = 70, i.e., 2x = 70 is our required linear equation in one variable.
All other word problems can be converted into mathematical statement or linear equations using above mentioned steps.