Maharashtra Board Class 6 Maths part 2 Chapter 10 Equations PDF Download

Equations

Teacher: Find two numbers and a mathematical operation to get the answer 15

Sharvari: 5 × 3 gives 15 and 45 divided by 3 also gives 15

Shubhankar: 17 - 2 gives 15 And 5 added to 10 also gives 15

Teacher: Very good! We see that the operations 5 × 3 and 17 - 2 both give the same result. We write this as 5 × 3 = 17 - 2. In mathematics, the sign of equality (=) shows that the numbers on both its sides are equal. They may be the result of different operations on the left and right hand sides. Such an expression of equality is called an equation.

Sharvari: Can we also write the equation 17 - 2 = 5 × 3?

Teacher: Yes, that equation is right, too. If you write a new equation simply by exchanging the two sides of an equation, then the new equation is also correct, that is, balanced.

If there are equal weights in both pans of a weighing scale, then the scale is balanced. Such a balanced scale is like an equation.

Teacher's Note

An equation is like a balance scale. When both sides are equal, it is balanced. Just like your parents balance money in home account every month.

Exam Trick

Remember: In an equation, the left side = right side, always. If you see 5 × 3 = 15, both sides give same answer 15.

Points to Remember

An equation shows that two sides are equal using the = sign.
Different operations can give the same result.
A balanced scale is like an equation with both sides equal.
We can exchange the two sides of an equation and it stays correct.

Different mathematical operations are given in the two rows below. Find out the number you get in each operation and make equations.

16 ÷ 2, 5 × 2, 9 + 4, 72 ÷ 3, 4 + 5

8 × 3, 19 - 10, 10 - 2, 37 - 27, 6 + 7

The Solution of an Equation

In the picture above, the distance between the house and the school is seen to be 300 m. On the same straight road, there is a shop between the school and the house. The distance between the shop and the house is 190 m. What is the distance between the school and the shop?

Use of a Letter for a Number

Teacher: See how the given information is shown in the picture above.

Sujata: Sir, why is the distance from the shop to the school shown as x?

Teacher: Instead of writing the number, we suppose that the distance is x. That is the distance we have to find out. Till we do so, we write it as x.

Samir: Then the sum of x and 190 should be 300.

Teacher: That's right! Let's write this in the form of an equation. Remember that x is a number but we do not know its value as yet.

\(x + 190 = 300\) What is the value of x here?

Shabana tried out various numbers for x. First she supposed x was equal to 70. The left hand side became 70 + 190 = 260. That was less than the right hand side. Then she took 150 for x and the left hand side became 340. This was greater than the right hand side. Finally, she chose 110 for the value of x. That made the left hand side the same as the right hand side and the equation was balanced. It meant that the value of x or the distance between the shop and the school was 110 metres.

In an equation, a letter is sometimes used in place of a number. A value for the letter has to be found that will make the equation balanced. Such a letter is called a variable.

The value of the variable which balances or satisfies the equation is called the solution to the equation. To solve an equation is to find the value of the variable in the equation or to find the solution to the equation.

In the example above, the solution to the equation '\(x + 190 = 300\)' is 110.

Teacher's Note

A variable is a letter that stands for a number we don't know. Like in a mystery, we use x to find the secret number. Just like you use a box to hide toys at home.

Exam Trick

Remember: Variable = unknown number. We use letters like x, y, a to show numbers we need to find. Think of it like a missing number in a puzzle.

Points to Remember

A variable is a letter used in place of an unknown number.
We use letters like x, y, and a to represent variables.
The solution is the value that makes the equation balanced.
To solve means to find the value of the variable.
The variable can represent any number that balances the equation.

Solving an Equation

Teacher: How can we find the weight of a guava in terms of bors?

John: If we remove 3 bors from each of the pans, they remain balanced, and then we can see that one guava weighs 4 bors.

Teacher: Excellent! You found the right operation. When solving an equation with one variable, we carry out the same operations on both sides of the equation to obtain simpler balanced equations, because, if the first equation is balanced then the new one obtained in this way is also balanced. The equations become simpler and simpler and finally we get the value of the variable, that is, the solution to the equation.

\(x + 3 = 7\)

\(\therefore x + 3 - 3 = 7 - 3\) (Subtracting 3 from both sides)

\(\therefore x + 0 = 4\)

\(\therefore x = 4\)

Let us take a second look at the previous equation.

\(x + 190 = 300\)

\(\therefore x + 190 - 190 = 300 - 190\) (Subtracting 190 from both sides)

\(\therefore x + 0 = 110\)

\(\therefore x = 110\)

While solving an equation, we can use this simple and unerring way rather than examining several random solutions.

Let us solve some examples using equations.

Example: Four years ago, Diljit was 8 years old. How old is he today?

Let us suppose he is a years old today.

Now, let's write the given information using a.

\(a - 4 = 8\)

\(\therefore a - 4 + 4 = 8 + 4\) (Adding 4 to both sides)

\(\therefore a + 0 = 12\)

\(\therefore a = 12\)

\(\therefore\) Diljit is 12 years old today.

Teacher's Note

To solve an equation, we do the same operation on both sides. This keeps it balanced, like a seesaw. If you add weight on one side, add the same on the other side too.

Exam Trick

Remember: Do the same operation on BOTH sides. If you subtract 3 from left, subtract 3 from right. This keeps the equation balanced like a scale.

Points to Remember

Do the same operation on both sides of an equation.
The equation stays balanced when we do the same thing to both sides.
Make the equation simpler and simpler until you get the answer.
Subtract if there is adding, add if there is subtracting.
The variable's value is the solution when the equation is balanced.

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