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).

Solving Linear Equation by Elimination Method

 Elimination method is a way to manipulate systems of equations in order to solve them algebraically. This is actually very similar to the way we manipulate single equations in order to solve them.

The elimination method is a technique for solving systems of linear equations. Let's walk through a couple of examples.


We're asked to solve this system of equations:
3y + 11x = −10
4y − 11x​​​ = 17
We notice that the first equation has a 11x and the second equation has a −11x term. These terms will cancel if we add the equations together—that is, we'll eliminate the x terms:​​
3y + 11x = −10
4y − 11x​​​ = 17
-----------------
7y + 0 = 7

Solving for y, we get:
7y + 0 = 7
7y = 7
y=1
plugging this value back into our first equation, we solve for the other variable:
3y + 11x = −10
3.1 + 11x = -10
11x = -10-3
11x = -13
x = -13/11
The solution to the system is x=−13/11 and y=1

Not all time this system will work easily. Suppose a system like 4x +2y = 24 and −6x +2y = 4. If you add these two equations together, no variables are eliminated. This time we have to multiply any equation with -1.

4x   + 2y    =    24 

 − (− 6x + 2y ) = −(4)
or
4x   +  2y   = 24  

6x  −  2y    = −4
-----------------------
10x + 0y =20

x = 2

Put this value back into our first equation, we solve for the other variable:

4x + 2y =24

4.2 + 2y = 24

8 + 2y = 24 

2y = 24-8

2y = 16

y = 8

The solution to the system is x=2 and y=8

What is an equation?

What do you notice about each of the following?
11 + 3 = 15
25 - 5 = 20
The right hand side is equal to left hand side.
An equation is a statement that two expressions are equal. For example, the expression 12+3 is equal to the expression 16-1(because they both equal 15), so we can write the following equation:
12+3=16-1

When the left-hand side was equal to the expression on the right-hand side then the equation is called true equations. Let's make sure we understand what a true equation is.
8+2 = 5+5
7-3= 4+1
Here one thing must be clear to you that 5+3 is an expression and it is numerical expression but if the expression includes variable then it is algebraic equation. For example, the equation x+2=6 has a variable in it.

How can we solve algebraic equation?

3 + Z =  17

if Z=14, then 3 + 14 = 17.

How to Convert from Binary to Decimal

Suppose a binary number 1100₂  and we will convert it into decimal form. The binary number has base 2. In the number (1100)₂ , 0 is the least significant digit and 1 is most significant digit. We will multiply each digit with the base 2 to the power index number. So binary number (1100)₂ has following index number of each digit- 

3 2 1 0 ---- index number or order of digit. It starts from 0
1 1 0 0 ---- the number

So, the multiplication will be like:

1100  =  1×2³  +  1×2²  +  0×2¹  +  0 × 2⁰  = 8 + 4 + 0 + 0 = 12 

So,  the decimal value of binary number (1100)₂ is  (12)₁₀

The fraction number can also be done in similar way but that time the power will be negative. Suppose the number (0.011)₂ .

0.011 = 0×2^-1  +  1×2^-2  +  1×2^-3    = 0 + .25 + .125  = 0.375 

So,  the decimal value of binary number (0.011)₂ is  (0.375)₁₀