Functions and Equations

An equation is a statement where two expressions are equal. Here left and right side will be equal. 

Suppose x + 3 = 10. For a particular value x, the equation will be satisfied. If x=7 then 7+3=10. The left and right side are equal.

Function is a bit different. It is a relationship between two variables. Function is written as f(x)=x+3. It takes a set of value as input and return a particular output. The input variable f(x) is called Independent variable because it takes input as f(0),f(1),f(2) etc. and returns corresponding output as 3,4,5 etc. x + 3 is dependent on f(x) that's why x+3 is called as Dependent variable.

In equation, we need a particular value of variable to make it equal where function takes a set of  permissible input and returns regarding output.

What is Function in Mathematics?

A function in mathematics is just like relation between input and output. So a function takes input and returns an output. It is denoted as f.  Most of the time, you see it in below form.

f(x) = x² + 1 

For a given value of x, you will get the result of  x² + 1.

Value of x	Output
0 1 2 3 4	1 2 5 10 17

We see that for a particular value of variable x, we get a corresponding output. This phenomena is called function in mathematics.

We shall denote function of x as f(x), F(x), φ(x) etc.

The number x that we use for the input of the function is called the 'Argument' of the function. It may seems to us that we can pick any number as argument  and for this reason we can call it 'Independent Variable' and the output of the function, e.g. f(x), f(3), etc. depends upon the argument, can be called the 'Dependent  variable'.
We can draw a picture of a function on graph by taking argument-value pairs of the function and describing each by a point in the plane, with x coordinate given by the argument and y coordinate given by the value for that pair.

Word Problem of System of Equation

Q. Jimmy is 12 years older than Brandon. 17 years ago, Jimmy was 4 times as old as Brandon. How old is Brandon now?

Suppose Jimmy's current age is J and Brandon's age is B. Jimmy is 12 years older than Brandon then
J = B + 12
17 years ago Jimmy was 4 times as old as Brandon.
So, 17 years ago Jimmy's age was J-12 and Brandon's age was B-17 then
J-17 = 4(B - 17)
Put J = B + 12 in above equation.
B + 12 - 17 = 4( B - 17 )
B - 5 = 4B - 68
3B = 63
B = 21
Brandon is 21 years old.

Q. Your class has 40 students and some want to watch movie and some want to watch stage drama. The cost of movie ticket is 20$ and stage drama is 10$. Total cost of ticket is 500$. How many students went to watch movie and how many want to watch stage drama.

M= Number of student to watch movie
S = Number of student to watch stage drama

We know that total student is 40. Then
M+S = 40 
We also know total cost of ticket is 500$. Then
20M + 10S = 500
Put M = 40 - S  in above equation, it will become
20(40 - S) + 10S = 500
800 - 20S + 10S= 500
800 - 10S = 500
10S= 300
S= 30

Put this value in M+S = 40 then M + 30 = 40.
M = 10
30 students went to watch stage drama and 10 students went to watch movie.

Various Forms of Linear Equations(Two Variables)

There are three major forms of linear equations: point-slope form, slope-intercept form and standard form.

Slope-intercept form: 
y = mx + b
where m is slope and b is y-intercept.
Example: y = 2x + 1
Slope: m = 2
Intercept: b = 1

Point-slope form:
y − y₁ = m(x − x₁)
where m is slope and (x₁,y₁) is a point on the line.
Example: y - 1= 2 (x - 3)
Slope: m = 2
x₁ = 1
y₁ = 3

Standard form:
Ax + By = C
Where A,B,C are constants. 
Example: y + 3x= -10
A = 1
B =3
C = -10

Number of solutions to a system of equations

System of Linear Equations:
A system of linear equations or system of equations means two or more linear equations that are being solved simultaneously.

What is the solution of a system of equations?

When two equations meet or intersect at a point then this point is called solution of the system.

A system of linear equations usually has a single solution, but sometimes it can have no solution (parallel lines) or infinite solutions (same line). This article reviews all three cases.

One solution:
This is the most common situation. Here, two lines meet at one point.
6x − 2y = 8​
4x + y = −1​​

Since the lines intersect exactly once, the system has exactly one solution.

No solution:
A system of linear equations has no solution when the graphs are parallel.
y = −x + 6​
5x + 5y = 15​​

We see two distinct parallel lines. We can confirm that the lines are indeed parallel, since the slopes of both lines are equal to −1. Since distinct parallel lines don't intersect, we conclude that the system has no solutions. 

Infinite solutions:
A system of linear equations has infinite solutions when the graphs are the exact same line.
​​​​​2x + y = 5​
14x + 7y = 35

The two lines are the same, they intersect infinitely many times. This means that the system has infinitely many solutions.

Solving Linear Equation by Substitution Method

The substitution method is another technique for solving systems of linear equations.

We're asked to solve this system of equations:
2y - 4x = 2
y -x = 4

Now we will find the value of x in second equation:
x = y - 4

Put the value of x in first equation.
2y - 4(y - 4) = 2
2y - 4y + 16 = 2
-2y = -16 +2
-2y = -14
y = 7

Now, put the value y=7 in second equation and we will get
7 - x = 4
x = 3


The solution of the linear system is (3, 7).