Linear Equations in One Variable

TOPICS What is Linear Equation? Application of Linear Equation Example 1 Example 2 Example 3 Equations Reducible to the Linear Form Assessment

What is Linear Equation? A statement of equality of two algebraic expressions in or more variables is called an equation. Every equation has a Left hand side(LHS), the equality sign ‘=’ and the Right hand side(RHS). The value of x, i.e. some number for x, which makes the equation a true statement is called solution or root of the equation.

Application of Linear Equation Linear equations can describe physics, business and biology. Systems of linear equations model phenomena with multiple relationships. They are a conceptual foundation for calculus-based theories of slope and are utilized in numeric approximations. Physics Equations of the form y = mx + b can track movement. The constant b defines the starting position, and m describes steady velocity. The variable x is the time during which movement is occurring at velocity m. Together, this information gives the total distance covered y, which is in the form of linear equation. Business In finance, if you start with $5 (b = $5) and work for 6 hours at $10/hr (m = $10/hr, x = 6 hours), after those 6 hours there is mx + b = 10(6) + 5 = $65.

Let the two consecutive numbers be x and x+1. Example 1: The sum of two consecutive numbers is 25. Find the numbers. Solution: Let the two consecutive numbers be x and x+1. So we can set up the following linear equation: Given that x + x+1= 25, 2x = 24, x = 24/2 = 12 So the other number is x + 1 = 12 + 1 = 13 Therefore, the two numbers are 12 and 13

x + 10 = 15,  x = 5. Example 2 Solution: Rakesh is 15 years now. He is 10 years older than his brother Mukesh. How old is Mukesh 10 years from now? Solution: Let Mukesh be x years old now. So, we set up the linear equation:  x + 10 = 15,  x = 5.  So, Mukesh is 5 years old now.  10 years from now, Mukesh will be 10 + 5 = 15

So, x + 20 = 3x 3x – x = 20, 2x = 20, x = 10 Example 3 Solution: 20 years from now, Suman will become three times as old as she is now. Find her age now Solution: Let Suman be x years old now.  20 years from now, she will be x +20 But 20 years from now, she we thrice her present age, x i.e. 3x So,  x + 20 = 3x 3x – x = 20, 2x = 20,  x = 10 Hence, Suman is 10 years old now. 

Equations Reducible to the Linear Form Solve the Linear Equation   Solution:

Assessment Present ages of Anu and Raj are in the ratio 4:5. Eight years from now the ratio of their ages will be 5:6. Find their present ages. Sum of the digits of a two-digit number is 9. When we interchange the digits, it is found that the resulting new number is greater than the original number by 27. What is the two-digit number? There is a narrow rectangular plot, reserved for a school, in Mahuli village. The length and breadth of the plot are in the ratio 11:4.At the rate Rs100 per metre it will cost the village panchayat Rs 75000 to fence the plot. What are the dimensions of the plot?

Simplify and solve the following linear equations.