ICSE Class 6 Maths Chapter 14 Substitution

Chapter 14 - Substitution

Including Use of Brackets as Grouping Symbols

14.1 Basic Concept

The value of an expression depends on the value(s) of its variable(s).

Consider the algebraic expression: \(3x + 2\)

In the expression \(3x + 2\), the variable used is x, and so the value of the expression \(3x + 2\) depends on the value of its variable x. That is:

(i) if \(x = 2\), the value of the expression \(3x + 2 = 3 \times 2 + 2 = 6 + 2 = 8\)

(ii) if \(x = 0\), the value of the expression \(3x + 2 = 3 \times 0 + 2 = 0 + 2 = 2\)

(iii) if \(x = -2\), the value of the expression \(3x + 2 = 3 \times -2 + 2 = -6 + 2 = -4\) and so on.

Now consider the algebraic expression \(5x - 2y\).

This expression consists of variables x and y.

Now, if:

(i) \(x = 3\) and \(y = 2\), \(5x - 2y = 5 \times 3 - 2 \times 2 = 15 - 4 = 11\)

(ii) \(x = 8\) and \(y = 5\), \(5x - 2y = 5 \times 8 - 2 \times 5 = 40 - 10 = 30\).

The process of finding the value of an algebraic expression by substituting the given value (values) of its variable (variables) is called substitution.

Example 1

If \(x = 6\) and \(y = 3\), find the value of:

(i) \(4x + y\)

(ii) \(3x - 4y\)

(iii) \(3xy\)

(iv) \(\frac{5x}{4y}\)

Solution

(i) \(4x + y = 4 \times 6 + 3 = 24 + 3 = 27\)

(ii) \(3x - 4y = 3 \times 6 - 4 \times 3 = 18 - 12 = 6\)

(iii) \(3xy = 3 \times 6 \times 3 = 54\)

(iv) \(\frac{5x}{4y} = \frac{5 \times 6}{4 \times 3} = \frac{30}{12} = \frac{5}{2} = 2\frac{1}{2}\)

Example 2

If \(a = 2\), \(b = 5\) and \(c = 8\), find the value of: \(3ab + 10bc - 2abc\)

Solution

\(3ab + 10bc - 2abc = 3 \times 2 \times 5 + 10 \times 5 \times 8 - 2 \times 2 \times 5 \times 8\)

\(= 30 + 400 - 160\)

\(= 430 - 160 = 270\)

Teacher's Note

Substitution is used in real life when you calculate costs, like finding the total price of groceries when you know the price per item and the quantity needed.

Example 3

If \(p = 8\), \(q = 1\) and \(r = 2\), find the value of: \(\frac{10pq - 3qr}{4pqr - 2p}\)

Solution

\(\frac{10pq - 3qr}{4pqr - 2p} = \frac{10 \times 8 \times 1 - 3 \times 1 \times 2}{4 \times 8 \times 1 \times 2 - 2 \times 8} = \frac{80 - 6}{64 - 16} = \frac{74}{48} = \frac{37}{24} = 1\frac{13}{24}\)

Example 4

If \(x = 2\); find the value of \(3x^2 + x\).

Solution

\(3x^2 + x = 3(2)^2 + 2\)

\(= 3 \times 2 \times 2 + 2 = 12 + 2 = 14\)

Example 5

If \(x = 5\), \(y = 6\) and \(z = 10\), find the value of:

(i) \(\frac{3x^2}{x}\)

(ii) \(\frac{xy}{xz}\)

(iii) \(\frac{x^2y}{z}\)

Solution

(i) \(\frac{3x^2}{x} = \frac{3 \times 5^2}{5} = \frac{3 \times 5 \times 5}{5} = 3 \times 5 = 15\)

Alternative method:

\(\frac{3x^2}{x} = \frac{3 \times x \times x}{x} = 3 \times x = 3 \times 5 = 15\)

(ii) \(\frac{xy}{xz} = \frac{5 \times 6}{5 \times 10} = \frac{30}{50} = \frac{3}{5}\)

Alternative method:

\(\frac{xy}{xz} = \frac{y}{z} = \frac{6}{10} = \frac{3}{5}\)

(iii) \(\frac{x^2y}{z} = \frac{5^2 \times 6}{10} = \frac{5 \times 5 \times 6}{10} = \frac{150}{10} = 15\)

Example 6

If \(a = 2\), \(b = 3\) and \(c = 4\), find the value of:

(i) \(a^b\)

(ii) \(b^a\)

(iii) \(c^b\)

(iv) \(a^2 - b^2 + c^2\)

Solution

(i) \(a^b = 2^3 = 2 \times 2 \times 2 = 8\)

(ii) \(b^a = 3^2 = 3 \times 3 = 9\)

(iii) \(c^b = 4^3 = 4 \times 4 \times 4 = 64\)

(iv) \(a^2 - b^2 + c^2 = (2)^2 - (3)^2 + (4)^2\)

\(= 4 - 9 + 16 = 20 - 9 = 11\)

Teacher's Note

Understanding exponents through substitution helps you calculate compound interest or population growth in exponential models used by scientists and economists.

Exercise 14(A)

1. Fill in the following blanks, when: \(x = 3\), \(y = 6\), \(z = 18\), \(a = 2\), \(b = 8\), \(c = 32\) and \(d = 0\).

2. Find the value of:

(i) \(p + 2q + 3r\), when \(p = 1\), \(q = 5\) and \(r = 2\)

(ii) \(2a + 4b + 5c\), when \(a = 5\), \(b = 10\) and \(c = 20\)

(iii) \(3a - 2b\), when \(a = 8\) and \(b = 10\)

(iv) \(5x + 3y - 6z\), when \(x = 3\), \(y = 5\) and \(z = 4\)

(v) \(2p - 3q + 4r - 8s\), when \(p = 10\), \(q = 8\), \(r = 6\), and \(s = 2\)

(vi) \(6m - 2n - 5p - 3q\), when \(m = 20\), \(n = 10\), \(p = 2\) and \(q = 9\)

3. Find the value of:

(i) \(4pq \times 2r\), when \(p = 5\), \(q = 3\) and \(r = 1/2\)

(ii) \(\frac{yx}{z}\), when \(x = 8\), \(y = 4\) and \(z = 16\)

(iii) \(\frac{a + b - c}{2a}\), when \(a = 5\), \(b = 7\) and \(c = 2\)

4. If \(a = 3\), \(b = 0\), \(c = 2\) and \(d = 1\), find the value of:

(i) \(3a + 2b - 6c + 4d\)

(ii) \(6a - 3b - 4c - 2d\)

(iii) \(ab - bc + cd - da\)

(iv) \(abc - bcd + cda\)

(v) \(a^2 + 2b^2 - 3c^2\)

(vi) \(a^2 + b^2 - c^2 + d^2\)

(vii) \(2a^2 - 3b^2 + 4c^2 - 5d^2\)

5. Find the value of: \(5x^2 - 3x + 2\), when \(x = 2\).

6. Find the value of: \(3x^3 - 4x^2 + 5x - 6\), when \(x = -1\).

7. Show that the value of: \(x^3 - 8x^2 + 12x - 5\) is zero, when \(x = 1\).

8. State true and false:

(i) The value of \(x + 5 = 6\), when \(x = 1\)

(ii) The value of \(2x - 3 = 1\), when \(x = 0\)

(iii) \(\frac{2x - 4}{x + 1} = -1\), when \(x = 1\).

9. If \(x = 2\), \(y = 5\) and \(z = 4\), find the value of each of the following:

(i) \(\frac{x}{2x^2}\)

(ii) \(\frac{xz}{yz}\)

(iii) \(z^x\)

(iv) \(y^x\)

(v) \(\frac{x^2 y^2 z^2}{xz}\)

(vi) \(\frac{5x^4 y^2 z^2}{2x^2}\)

(vii) \(xy \div y^2z\)

(viii) \(\frac{x^2 y^x}{x}\)

10. If \(a = 3\), find the values of \(a^2\) and \(2^a\).

11. If \(m = 2\), find the difference between the values of \(4m^3\) and \(3m^4\).

Teacher's Note

Substitution exercises strengthen mental math skills used in budgeting, cooking adjustments, and scientific calculations in laboratories.

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