Ch 4: Linear Equations - Exercise 4.2 Practice

Q1: Number of Solutions

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Which one of the following options is true, and why?
The equation $y = 7x - 2$ has:
(i) a unique solution,
(ii) only two solutions,
(iii) infinitely many solutions

(iii) Infinitely many solutions.

Reason: A linear equation in two variables has infinitely many solutions.

For every value of $x$, there is a corresponding value of $y$, and vice versa.

Infinitely many solutions

Q2: Find Solutions

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Write four solutions for each of the following equations:
(i) $3x + y = 9$
(ii) $x = 5y$
(iii) $2x + 3y = 12$

(i) $3x + y = 9$:
If $x=0, y=9$. $(0, 9)$
If $x=1, y=6$. $(1, 6)$
If $x=2, y=3$. $(2, 3)$
If $x=3, y=0$. $(3, 0)$

(ii) $x = 5y$:
If $y=0, x=0$. $(0, 0)$
If $y=1, x=5$. $(5, 1)$
If $y=2, x=10$. $(10, 2)$
If $y=-1, x=-5$. $(-5, -1)$

(iii) $2x + 3y = 12$:
If $x=0, y=4$. $(0, 4)$
If $x=3, y=2$. $(3, 2)$
If $x=6, y=0$. $(6, 0)$
If $y=6, x=-3$. $(-3, 6)$

Q3: Check Solutions

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Check which of the following are solutions of the equation $2x - y = 4$ and which are not:
(i) $(2, 0)$
(ii) $(0, -4)$
(iii) $(3, 2)$
(iv) $(1, 1)$
(v) $(2, 2)$

Equation: $LHS = 2x - y$, $RHS = 4$.

(i) $(2, 0)$: $2(2) - 0 = 4$. (Yes)

(ii) $(0, -4)$: $2(0) - (-4) = 4$. (Yes)

(iii) $(3, 2)$: $2(3) - 2 = 6 - 2 = 4$. (Yes)

(iv) $(1, 1)$: $2(1) - 1 = 1 \neq 4$. (No)

(v) $(2, 2)$: $2(2) - 2 = 2 \neq 4$. (No)

(i), (ii), and (iii) are solutions.

Q4: Find k

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Find the value of $k$, if $x = 2, y = 3$ is a solution of the equation $3x + 4y = k$.

Substitute $x = 2$ and $y = 3$ in the equation $3x + 4y = k$.

$3(2) + 4(3) = k$

$6 + 12 = k$

$k = 18$

k = 18