# Algebraic substitution 2

Last updated: September 24, 2021

## What is substitution?

The basic idea behind substitution is how we can put values of variables in the equation to find the values of other variables.

We can readily comprehend the notion of substitution with the help of an example; imagine a person with various notes in varying numbers in his wallet. To find the total amount he has, we can form equations, like he has $$x$$ notes of $$10$$ each, $$y$$ notes of $$50$$ each, so we can develop the equation as $$z=10x+50y$$.

## E2.1: Substitute numbers for words and letters in complicated formulae.

Solving an equation by the substitution method is the easiest method. No matter how complicated or simple the equation is, but substituting solves it without difficulty. In this method, we are first given the equation that needs to be solved, then, the values that we need to be substituted and then we are asked to find the solution of the equation in whatever way is required. This could be explained with an example,

Take an equation:

Say, $$2x+ 4y=7z$$

Here, it is given that value of $$x$$ is $$4$$ and the value of $$y$$ is $$2$$.

After putting the values into the equation, we can then find the value of $$z$$.

Other questions can also be solved as per using the given method.

### Worked examples of substitution questions

Example 1: Suppose a boy was playing with a football on the field. The radius of the ball is $$7\;\text{cm}$$. He has to find the surface area and volume of the ball. Help him to find the surface area and volume by substituting the value of the radius of the ball. The formulas of surface area and volume of the sphere are as follows.

• The surface area of the sphere is $$4\pi r^{2}$$.

• Volume of the sphere is $$\frac{4}{3}\pi r^{3}$$.

Question 1

Step 1: Write the given values.

The radius of the ball is given $$r=7\;\text{cm}$$

Step 2: Recall the formula for the surface area of the sphere.

The surface area of the sphere is $$4\pi r^{2}$$.

Step 3: Substitute the value of $$r$$ and $$\pi$$ in the given formula.

$$\text{Surface area of sphere}=4\pi r^{2}$$

$$4\times \pi \times 7^{2}$$

Step 4: Solve further to get the final answer.

$$\text{Surface area of sphere}=4\times \pi \times 7 \times 7$$

$$4\times \pi \times 49$$

$$196 \pi$$

Step 5: Write the surface area of the sphere.

The surface area of the given sphere is $$196 \pi\;\text{cm}^2$$.

Question 2

Step 1: Recall the formula for the volume of the sphere.

The volume of sphere is $$\frac{4}{3}\pi r^{3}$$.

Step 2: Substitute the value of $$r$$ and $$\pi$$ in the given formula.

$$\text{Volume of the sphere}=\frac{4}{3}\pi r^{3}$$

$$\frac{4}{3}\times \pi \times 7^{3}$$

Step 3: Solve further to get the final answer.

$$\text{Volume of the sphere}=\frac{4}{3}\times \pi \times 7 \times 7\times 7$$

$$\frac{4}{3}\times \pi \times 343$$

$$457 \frac{1}{3} \pi$$

Step 4: Write the Volume of the sphere.

The volume of the given sphere is $$457 \frac{1}{3} \pi\;\text{cm}^3$$

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