Linear Equations

Linear Equations

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Linear Equation: Definition

A mathematical statement that has an equal to "=" symbol is called an equation.
 
Linear equations are equations with degree 1

Understanding Linear Equation

Appu created a game called "Mind Reader".

He wants to play it with his friends.

He asks one of his friends, Kaira, to think of a number, multiply it by 2 and subtract 5 from it.

He asks her the final result.

Kaira says, "it is 13"

Inroducing linear equation with a help of a example: A boy introduces a game Mind Reader to his friends.

Appu instantly says that the number Kaira thought of initially was 9

Kaira nods and Appu's friends including Kaira are surprised!

Everybody wants to know how the game "Mind Reader" works.

Do you know how it works?

By the end of this short lesson, you will understand how it works.


Linear Equation: Formula

The standard form of a linear equation with variables \(x\) and \(y\) is: 
\(Ax+By=C\)
The standard form of a linear equation is Ax+By=C.

Linear Equation: Examples

A linear equation can be written in different ways.

Look at the following equations.

  Equations Linear or Non-Linear
1. \(y=8x-9\) Linear
2. \(y=x^2-7\) Non-Linear
3. \(\frac{x}{5}=4\) Linear
4. \(\sqrt{y}+x=6\) Non-Linear
5. \(y+3x-1=0\) Linear
6. \(y^3-x=9\) Non-Linear

Linear Equations on Graph

The graph of a linear equation in one variable \(x\) forms a vertical line parallel to \(y\)-axis and vice-versa.

The graph of a linear equation in two variables \(x\) and \(y\) forms a straight line.

The graphical representation of linear equations with one and two variables.

The reason an equation of degree one is called a linear equation is that its geometrical representation is a straight line.

Explore the graphs of linear equations using the simulation below.


How to Solve Linear Equations?

An equation is like a weighing balance with equal weights on both sides.

Explaining the way to solve a linear equation by weighing balance.

If we add or subtract the same number from both sides of an equation, it still holds.

Similarly, if we multiply or divide the same number on both sides of an equation, it still holds.

Consider the equation, \(3x-2=4\)

The LHS and RHS of the linear equation 3x-2=4 are shown on a weighing balance

We will perform mathematical operations on the LHS and the RHS so that the balance is not disturbed.

Let's add 2 on both sides to reduce the LHS to \(3x\)

This will not disturb the balance.

The new LHS is \(3x-2+2=3x\) and the new RHS is \(4+2=6\)

Solving the linear equation 3x-2=4 on weighing balance.

Now let's divide both sides by 3 to reduce the LHS to \(x\)

Thus, we have \(x=2\)

Solution of linear equation 3x-2=4 on a weighing balance.

 
Thinking out of the box
Think Tank
   
1. Solve the riddle:
  Riddle to solve a linear equation


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Linear Equation: Problems and Solutions

Example 1

 

 

A one-day International cricket match was organised in Nagpur.  

India and Sri Lanka were the two teams.

Two Indian batsmen together scored 189 runs. 

Example for a linear equation: Two batsmen score 189 runs together.

Can you express this information in the form of a linear equation?

Solution:

Let us use the variables \(x\) and \(y\) to denote the number of runs scored by each batsmen.

We know that they scored 189 runs together.

Thus, the total of \(x\) and \(y\) is 189

\(\therefore x+y=189\)
Example 2

 

 

Bansi loves to collect two-rupee and five-rupee coins in her piggy bank.

Example for a linear equation: A girl's piggy bank has a sum of Rs. 77 and has 3 times as many two-rupee coins as five-rupee coins in it.

She knows that the total sum in her piggy bank is Rs. 77 and it has 3 times as many two-rupee coins as five-rupee coins in it.

She wants to know the exact number of two-rupee coins and five-rupee coins in her piggy bank. 

Can you help her find the count?

Solution:

Let the number of five-rupee coins be \(x\).

The number of two-rupee coins will be \(3x\)

Total amount = \((5\times x)+(2 \times 3x)\)

Example to understand linear Equations using denominations of money

According to the information we have,

\[\begin{align}(5\times x)+(2 \times 3x)&=77\\5x+6x&=77\\11x&=77\\x&=7\end{align}\]

\(\therefore\) She has 7 five-rupee & 21 two-rupee coins

Linear Equation: Calculator

Try solving linear equations in one variable in the simulation below.

 
important notes to remember
Important Notes
  1. The values of the variable that makes a linear equation true are called the solution or root of the linear equation.
  2. The solution of a linear equation is unaffected if the same number is added, subtracted, multiplied or divided on both sides of the equation.
  3. The graph of a linear equation in one or two variables is a straight line.

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Practice Questions

Here are few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 

Maths Olympiad Sample Papers

IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.

You can download the FREE grade-wise sample papers from below:

To know more about the Maths Olympiad you can click here


Important Topics

However, before you dive into the deep end of the pool, it’s best to familiarize yourself with the basics. Check out the links below to find out everything that Cuemath has to offer on the topic of Linear and Quadratic equations. Below topics cover both concepts and worksheets:


Frequently Asked Questions (FAQs)

1. What is a linear equation? Explain with an example.

An equation of the form \(ax+by=c\) is called a linear equation. Here, \(x\) and \(y\) are variables and \(a,b\) and \(c\) are constants. 

Examples of linear equation are:

  1. \(y=4x-3\)
  2. \(7y-5x=1\)
  3. \(y=2\)

2. What is the formula for a linear equation?

The standard form of a linear equation in one variable is of the form \(ax+b=0\)

Here, \(x\) is a variable and \(a\) and \(b\) are constants. 

The standard form of a linear equation in two variables is of the form \(ax+by=c\)

Here, \(x\) and \(y\) are variables and \(a, b\) and \(c\) are constants.

3. What is a simple definition of a linear equation?

An equation that can be written in the form \(ax+by=c\) is called a linear equation.

This is the standard form of a linear equation in two variables \(x\) and \(y\)

  
More Important Topics
Numbers
Algebra
Geometry
Measurement
Money
Data
Trigonometry
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