Linear equation

A linear equation in one variable is an equation in which highest power of the variable is one.

In general the equation of the form ax + b =c , where a, b and c are rational numbers and a
=
0

Is called a linear equation in one variable ‘x’. The value of the variable which satisfy the given equation

is called a solution or the root of the equation.

The standerd form of the Linear equation in one variable is px + q = 0, where p and q are rational numbers q
=
0.

Its solution is given by x = -
pq
.


Rules for solving Linear Equations: -
Add the same number to both sides of the equation.
Subtract the same number from both sides of equation .
Multiply or divide both sides of the equation by the same non-zero number.
Transpose any term of the equation from one side to the other by changing its sign.



Remember that an equation does not change when the above rules are applied.


Solving Linear Equations: -


Solving equation of the form x + b = c



Let us consider a equation

x + 7 = 12



1. Subtract 7 from both sides of the equal sign:

x + 7 - 7 = 12 - 7

2. Simplify the result:

x = 5
Solving equation of the form x – b = c

Let us consider a equation

x – 13 = 11

1. Add 13 to both sides of the equal sign:

x - 13 + 13 = 11 + 13

2. Simplify the result:

x = 24
Solving equation of the form bx = c

Let us consider an example

10x = 50

1. Divide both sides of the equal sign by 10:

1010x=1050

2. Simplify the result:

x = 5
Solving equation of the form
xb=c

Let us consider an example

x7=10

1. Multiply both sides of the equal sign by 7:

77x=
(7)(10)

2. Simplify the result:

x = 70

That all we did to be able to solve the most complicated linear equation in one variable that is of the form
cx+dax+b=k
where a, b, c, b and k are numbers, x is the variable and
cx +d=0
by reducing it to linear equation .

Solution of the equation of the form
cx+dax+b=k

Cross Multiplication Method
In this method following steps are followed
Multiply the numerator of L.H.S ( left hand side of = ) by the denominator of R.H.S ( Right hand side of = ).
Multiply the numerator of R.H.S by the denominator of L.H.S.
Equate the expression obtained in (1) and (2).



Let us take an example.

Solve the equation \frac{7y-5}{4y+2}=\frac{8}{7} .

On Cross multiplying, we get,

7 ( 7y – 5 ) = 8 ( 4y + 2 )

or 49y – 35 = 32y + 16

49y – 32y = 16 + 35 [Transposing 32y to L.H.S. and –35 to R.H.S.]

or 17y = 51

or
y=1751=3


Application of linear equation in one variable: -

Strategy for Solving Verbal Problems

1. Read the problem carefully.

2. Use diagrams or charts if you think it will make the information clearer.

3. Find the relationship or formula relevant to the problem.

4. Identify the unknown quantity (or quantities), and label them, using one variable.

5. Write an equation involving the unknown quantity, using the relationship or formula from step 3.

6. Solve the equation.

7. Answer the question.

8. Check the answer in the original words of the problem.

Example. A total of 20000wasinvestedintwobondmutualfundsajunkbondfundandagovernmentbondfundThejunkbondfundisriskyandyields111300. How much was invested in each fund?

Solution. Let x = the amount of money invested in the junk bond fund

20,000 – x = the amount of money invested in the government bond fund

To calculate interest, we use the simple interest formula: interest = principle · rate · time

Since the time is one year, in this problem, interest = principle · rate

Amount invested

(Principal)
Interest

Rate
Interest Earned

Junk Bonds
x
0.11
0.11x

Government Bonds
20,000 – x
0.05
0.05(20,000 – x)

Total
20,000

1300


The total interest earned is the sum of the interest earned from each fund. We obtain the following equation from the "Interest Earned" column of the chart:
0.11x + 0.05(20,000 – x) = 1300


0.11x + 1000 – 0.05x = 1300
Distributive Property

0.06x + 1000 = 1300
Combine like terms

0.06x = 300
Subtract 1000 from each side


x = 5000
Divide both sides by 0.06


The amount invested in the junk bond fund, x, is $5000.

The amount invested in the government bond fund, 20000 – x, is $15,000.
Things to be remember: -
An equation is a statement of equality of two algebraic expressions involving an unknown quantity called a variable.
An equation in which the highest power of the variable is one, is called a linear equation.
Rules for solving a linear equation are:

a) Add the same number to both side of equation.

b) Subtract the same number from both side of equation.

c) Multiply or divide both sides of the equation by the same non-zero number.

d) Transpose any term of the equation from one side to the other by changing its sign.
4. If
cx+dax+b=k
,then 1(ax + b) = k(cx + d) is called a cross multiplication method.
To solve a word problem, denote the unknown quantity by some variable and translate the statement given in the problem step by step into a mathematical statement. Using the given condition, formulate an equation and solve the equation for unknown quantity.