# Algebraic substitution 1

Last updated: September 24, 2021

## What is algebraic substitution?

Substitution is the procedure of putting one thing like a number, a letter or symbol in place of other. In simple words, whenever we use numbers at the position of letters, it is called the substitution process.

Substitution is used to solve an equation, find the value of a given letter, and so on. That is the reason substitution plays an important role in understanding equations along with their variables.

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

Substitution means putting numbers in place of letters. Now, we shall learn the process of substitution with the help of some examples.

### Worked examples of algebraic substitution

Example 1: If it is given that $$a=4$$, $$b=2$$ and $$c=3$$, evaluate the following expressions.

1. $$2a+5b+6c$$

2. $$a-2b+4c$$

3. $$2b-3a-6c$$

4. $$3a+b-c$$

Step 1: Write the first expression.

$$2a+5b+6c$$ is the given expression.

Step 2: Solve the first expression substituting the values of $$a$$, $$b$$ and $$c$$.

$$2 \times 4+ 5 \times 2+ 6 \times 3$$

Step 3: Solve further to find the value of the above expression.

$$8 +10 +18= 36$$

Step 4: Write the final answer.

$$2a+5b+6c= 36$$

Step 5: Write the second expression.

$$a-2b+4c$$ is the given expression.

Step 6: Solve the second expression substituting the values of $$a$$, $$b$$ and $$c$$.

$$4- 2 \times 2+ 4 \times 3$$

Step 7: Solve further to find the value of the above expression.

$$4 -4 +12= 12$$

Step 8: Write the final answer.

$$a-2b+4c= 12$$

Step 9: Write the third expression.

$$2b-3a-6c$$ is the given expression.

Step 10: Solve the third expression substituting the values of $$a$$, $$b$$ and $$c$$.

$$2 \times 2 -3 \times 4- 6 \times 3$$

Step 11: Solve further to find the value of the above expression.

$$4-12-18= -28$$

Step 12: Write the final answer.

$$2b -3a -6c= -28$$

Step 13: Write the fourth expression.

$$3a+b-c$$ is the given expression.

Step 14: Solve the fourth expression substituting the values of $$a$$, $$b$$ and $$c$$.

$$3 \times 4 +2 -3$$

Step 15: Solve further to find the value of the above expression.

$$12+2-3= 11$$

Step 16: Write the final answer.

$$3a +b -c=11$$

Example 2: Evaluate the value of the variables $$x$$, $$y$$ and $$z$$. Substitute $$p=2$$, $$q=3$$ and $$r=4$$.

1. $$x=p^{2}\left(q-r\right )$$

2. $$y=q^{2}r-2pq$$

3. $$z=pq\left(q+r \right)+p^{2}$$

Step 1: Write the first equation.

$$x=p^{2}\left(q-r\right )$$ is the given equation.

Step 2: Find the value of $$x$$ substituting the values of $$p$$, $$q$$ and $$r$$ in the equation.

$$x= \left( 2\right)^{2}\left(3-4 \right)$$

Step 3: Solve further to find the value of $$x$$.

$$x=4 \times \left(-1 \right)$$

$$x=-4$$

Step 4: Write the final answer.

$$x=-4$$

Step 5: Write the second equation.

$$y=q^{2}r-2pq$$ is the given equation.

Step 6: Find the value of $$y$$ substituting the values of $$p$$, $$q$$ and $$r$$ in the equation.

$$y=\left(3 \right)^{2} \times 4 -\left(2\times 2\times 3 \right)$$

Step 7: Solve further to find the value of $$y$$.

$$y=9\times 4-12$$

$$y=36-12=24$$

Step 8: Write the final answer.

$$y= 24$$

Step 9: Write the third equation.

$$z=pq \left(q+r \right)+p^{2}$$ is the given equation.

Step 10: Find the value of $$z$$ substituting the values of $$p$$, $$q$$ and $$r$$ in the equation.

$$z=2\times 3\left(3+4 \right)+\left(2 \right)^{2}$$

Step 11: Solve further to find the value of $$z$$.

$$z=2\times 3\times 7+ 4$$

$$z=42+4=46$$

Step 12: Write the final answer.

$$z=46$$