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Linear Equations Formula

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A linear equation is known as the algebraic equation that represents the straight line. It is composed of variables and constants. Linear equations consist of the first-order, that involves the highest power to any of the involved variables i.e.

  • It is also considered as the polynomial of a degree
  • The equation that contains only one variable is known as the homogeneous equation.

That corresponding variable in an equation is known as the homogeneous variable.

For example, we can see as,

​2x = 3 (it is a linear equation in one variable)

x + 2y = 3 (it is a linear equation in two variables)

x + y + z = 8 (it is in three variables)

x + y² = 1 (this equation doesn’t come in the linear equation because the highest power of variable)

Linear Equation in One variable

Linear equation in one variable is expressed as the ax + b = 0 or ax = b, here the involved variable and constants are x, a, and b. The constants(a and b) in this equation must be a non-zero real number. These equations have only one type of solution to get the value of a variable (x)

Steps to solve the linear equation in one variable are as follows:

Step 1: In this type of equation the constants a and b are some fractional numbers, for which LCM has to be taken to clear them.

Step 2: All the constants in this equation should be taken to the right side of the equation.

Step 3: The terms that involve a variable should be kept on the left-hand side of the equation, this helps to evaluate the value of the variable. After this, the equation is verified and we get the answer.

Examples of linear equations in one variable

22x = 65

6/5 + 1/3 x = 2

8y – 3 = 7

4/3 (z – 2) = 0

After analyzing these examples we get to know that each one of the equation has only one variable in it and the highest power the variable get is 1. These algebraic equations can be solved by taking all the variables on the left-hand side (L.H.S) and constants on the right-hand side (R.H.S), to solve the corresponding variable value

Linear equation in Two Variable

Linear equation in two variables comes into a picture, this means that the whole equation has 2 variables presented in it. So, the linear equation in two variables can be expressed as,  ax + by + c = 0, where a, b, c are the constants and x, y are the variables.

Linear equation in two variables is given as

Ax + By + C = 0 where A, B and C are about real numbers and A and B can never be zero.

Even the linear equation in one variable can also be expressed as the linear equation in two variables

x.1 + y.0 = 3

Example: A one-day international match was played between South Africa and India in Nagpur. Two Indian batsmen scored a total of 158 runs. Express this information in the form of an equation.

Solution:

Here we know that two batsmen have scored a total of 158 runs but we don’t know that how much each batsman has scored. So, let’s assume that the runs scored by each batsman are x and y

So, the equation will be

x + y = 158

This is the linear equation in two variable

Solution of linear equation in Two Variable

We have seen some equations like x = 6, y = 12. This type of equation has only one solution. But when it comes to the linear equation in two variables then there can be more than one solution.

For example: Let’s assume an equation in two variables,

x + 3y = 6

Firstly to get the solution of this equation we have to get the values of the variables x and y which can satisfy the equation, in this equation x=3 and y=1. So, let’s verify the equation mentioned above

x + 3y = 6

(3) + 3(1) = 6

3 + 3 = 6

We can do more solutions for this equation like assuming the variable x and then putting the value in the equation. Like let’s assume x=6 now plug it in the equation we get

6 + 3y = 6

3y = 0

y = 0

As we take different values for the variable we will have infinite solutions for a particular equation.

Similar Problems

Question 1: Solve for y, 6y – 3 = 0

Solution:

Solving for the value of y,

Adding 3 to both sides of the equation,

⇒ 6y – 3 + 3 = 3

⇒ 6y = 3

Dividing both sides of the equation by 6

⇒ y = 3/6

Simplifying the equation,

⇒ y = 1/2

Question 2: Solve the equation in x, 4/5x -5 = 15

Solution:

4/5x -5 = 15

Taking constants to RHS,

4/5x= 15+5

4/5x = 20

x = 100/4

x = 25

Question 3: There are two numbers, one equal to 7/6 and the other equal to 1/3 times some number x. The sum of these two numbers is 1. Find x.

Solution:

The sum of both the numbers is 1 so the equation will be,

7/6 + 1/3x = 1

Taking all the constants to the R.H.S of the equation.

1/3x = 1 – 7/6

1/3x = -1/6

Multiplying both the side of the equation by 3

3 (1/3x) = 3 × (-1/6)

x = -1/3

Question 4: Solve the equation in x, 3x + 5y = 33, where y = 3

Solution:

We have been provided with an equation

3x+5y= 33

We have to find the value of x as the value of y is provided in the question

y= 3

So, putting the value of y in the equation

3x+5(3)= 33

3x+15= 33

By taking all the constants to the R.H.S of the equation.

3x = 33-15

3x= 18

x= 18/3

x = 6

So here the value of x is 6

Question 5: There are two numbers, one equal to 2/4 some number y, and the other equal to 1/3 times some number x. The sum of these two numbers is 3. Find y. And the value of x is x = 2

Solution:

The sum of both the numbers is 3 so the equation will be,

2/4y + 1/3x= 3

Pitting the value of x the equation will be

2/4y + 1/3(2)= 3

2/4y + 2/3= 3

Taking all the constants to the R.H.S of the equation.

2/4y = 3-2/3

2/4y = 4/3

y= 4*4/3*2

Y=16/6

y= 8/3

So, the value of y will be 8/3



Last Updated : 10 Jan, 2024
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