Linear Equation

What is linear equation?

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable. Linear equations can have one or more variables. 

Example:

• A linear equation in one unknown x may always be rewritten

• If a ≠ 0, there is a unique solution

• If a = 0, then either the equation does not have any solution, if b ≠ 0 (it is inconsistent, or every number is a solution, if b is also zero.

What is linear equation in one variable?

A linear equation in one variable is an equation which can be written in the form:

Ax + B = C, where A, B, and C

are real numbers, with A 0.

Important

Linear equations in one variable:

2x + 3 = 11 2(x 1) =

8

Not linear equations in one variable:2x + 3y = 11Two

variables

can be rewritten 2x + ( 2) = 8.

x is squared.

(x 1)2

= 8

7532

xx can be rewritten x + 5 = 7. 3

1

7532

xx

Variable in the denominator

To solve a linear equation

in one variable: 1. Simplify both sides of the equation.

2. Use the addition and subtraction properties to get all variable terms on the left-hand side& all constant terms on the right-hand side.

3. Simplify both sides of the equation. 4. Divide both sides of the equation by the coefficient of the variable.

Example :

Solve x + 1 = 3(x 5). x + 1 = 3(x 5) x + 1 = 3x 15 x = 3x 16 2x = 16

The solution is 8. Check the solution:

TRUE

(8) + 1 = 3((8) 5) 9 = 3(3)

Uses of

Linear Equati

on

In the Law

n

SnowEverywhe

re

In the Pool

SNOW

Suppose a water district wants to know how much snow melt runoff it can expect this year. The melt comes from a big valley, and every year the district measures the snowpack and the water supply. It gets 60 acre-feet from every 6 inches of snow pack.

CONTINUED:

This year surveyors measure 6 feet and 4 inches of snow. The district put that in the linear expression (60 acre-feet/6 inches) * 76 inches. Water officials can expect 760acre-feet of snowmelt from the water.

In the Pool Its springtime and

Puneet wants to fill his swimming pool. He doesn't want to stand there all day, but he doesn't want to waste water over the edge of the pool, either. He sees that it takes 25 minutes to raise the pool level by 4 inches.

Continued: He needs to fill the pool to a depth of 4 feet; he has 44 more inches to go. He figures out his linear equation: 44 inches * (25 minutes/4 inches) is 275 minutes, so he knows he has four hours and 35 minutes more to wait.

IN THE LAWN

Harshit has noticed that in it's springtime. The grass has been growing. It grew 2 inches in two weeks. He doesn't like the grass to be taller than 2 1/2 inches, but he doesn't like to cut it shorter than 1 3/4 inches.

Continued: How often does he need to cut the lawn? He just puts that calculation in his linear expression, where (14 days/2 inches) * 3/4 inch tells him he needs to cut his lawn every 5 1/4 days. He just ignores the 1/4 and figures he'll cut the lawn every five days.

Everywhere

It's not hard to see other similar situations. If you want to buy beer for the big party and you've got 60 Rupees in your pocket, a linear equation tells you how much you can afford.

Whether you need to bring in enough wood for the fire to burn overnight, calculate your, paycheck, figure out how much paint you need to redo the upstairs bedrooms or buy enough gas to make it to and from your Aunt Sylvia's, linear equations provide the answers. Linear systems are, literally, everywhere

CONTINUED: