Equation

Equation

In this mini-lesson, we will explore the concept of equations by knowing the definition of an equation, equation examples, equation of a circle, quadratic equation, linear equation, and types of equations.

Before we get started, let's see what Joe did.

Joe 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.

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

Kaira nods and Joe'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.

Lesson Plan


What are Equations?

Meaning of Equation in Maths

An equation is a mathematical statement with an 'equal to' symbol between two algebraic expressions that have equal values.

For example, \(2x+9\) is the expression on the left-hand side that is equated with the expression 24 on the right-hand side.

Example of an equation

Look at the following examples. These will give you an idea of the meaning of an equation in math.

  Equations Is it an equation?
1. \(y=8x-9\) Yes
2. \(y+x^2-7\) No, because there is no 'equal to' symbol.
3. \(7+2=10-1\) Yes

How Do You Solve 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 of a line, \(3x-2=4\)

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, the solution of the equation of a line is \(x=2\)

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

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

What Are the 3 Types of Equations?

There are 3 main types of equations in math.

Linear Equation

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.

Quadratic Equation

The standard form of a quadratic equation with variable \(x\) is:
\(ax^2+bx+c=0\), where \(a\neq 0\)

The standard form of a quadratic equation

Cubic Equation

The standard form of a cubic equation with variable \(x\) is: 

\(ax^3+bx^2+cx+d=0\), where \(a\neq 0\)

Standard form of a cubic equation

 
important notes to remember
Important Notes
  1. The values of the variable that makes an equation true are called the solution or root of the equation.
  2. The solution of an 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.
  4. The curve of the quadratic equation is in the form of a parabola.

Equation Calculator

Try solving linear equations in one variable using the equation calculator below.

You can also verify your answers in the end.


Solved Examples

Example 1

 

 

Hailey loves to collect pennies and dimes in her piggy bank.

Example for a linear equation: Hailey has money that equals 77 cents in her piggy bank.

She knows that the total sum in her piggy bank is Rs.100 and it has 3 times as many coins of Rs.5 as notes of Rs.10 in it.

She wants to know the exact number of coins of Rs.5 and the number of notes of Rs.10 in her piggy bank. 

Can you help her find the count?

Solution

Let the number of notes of Rs.10 be \(x\).

Then the number of coins of Rs.5 will be \(3x\)

Therefore, total amount = \((10\times x)+(5 \times 3x)=10x+15x=25x\)

According to the information we have,

\[\begin{align}25x&=100\\[0.2cm]x&=4\end{align}\]

Therefore, 

The number of notes of Rs.10 is, \(x=4\)
The number of cois of Rs.5 is, \(3x=12\)
   Example 2

 

 

Mia is a fitness enthusiast who goes running every morning.

The park where she jogs is rectangular in shape and measures \(12\;\text{metres}\) by \(8\;\text{metres}\).

Quadratic Equation Question to find width of a park's pathway

An environmentalist group plans to revamp the park and decides to build a pathway surrounding the park.

This would increase the total area to \(140\;\text{sq. metres.}\)

What will be the width of the pathway?

Solution

Let’s denote the width of the pathway as \(x\).

Then, the length and breadth of the outer rectangle is \((12+2x)\;\text{metres }\) and \((8+2x)\;\text{metres}\)

According to the question,
\[\begin{align}(12+2x)(8+2x)&=140\\2(6+x)\cdot 2(4+x)&=140\\(x+6)(x+4)&=35\\x^2+10x-11&=0\\x^2+11x-x-11&=0\\x(x+11)-(x+11)&=0\\(x+11)(x-1)&=0\\x&=1,-11\end{align}\]

Since length can’t be negative, we take \(x=1\)

\(\therefore\) Width of the pathway  \(= 1\;\text{metre}\)
Example 3

 

 

Meghan will begin attending classes for the sixth grade from next week.

He suddenly realized that he doesn't have either a notebook or a pen.

He goes to a shop to purchase them and realizes that the cost of a notebook is Rs.5 more than twice the cost of the notebook.

Linear Equations: Carlos wants to buy a notebook and a pen

Represent this above information using equations in two variables.

Solution

Let's assume the following.

The cost of a pen = Rs.\(x\)

The cost of a notebook = Rs.\(y\)

According to the given information,

\(\therefore\) \(y = 2x + 5\)

Interactive Questions on

Here are a few activities for you to practice.

Select/type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 

Let's Summarize

The mini-lesson targeted the fascinating concept of the equation. The math journey around equation starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.


Frequently Asked Questions (FAQs)

1. What is a true equation?

If the values on both sides of the 'equal to' symbol hold true, then the equation is a true equation.

2. What is the equation of a circle?

The equation of a circle with radius \(r\) and center \((x_{1}, y_{1})\) is \((x-x_{1})^2+(y-y_{1})^2=r^2\).

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