3 Marks Question

Question 13 Marks

Solve the equation $2y - 1 = y + 1$ and represent it graphically on the coordinate plane.

Answer

We are given,
$2y - 1 = y + 1$
we get,
$2y - y = 1 + 1$
$y = 2$
The representation of the solution on the Cartesian plane, it is a line parallel to y axis passing through the point $(0, 2)$ is shown below

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Question 23 Marks

Write two solutions of the form x = 0, y = a and x = b, y = 0 for the following equations: 5x - 2y = 10

Answer

We are given, 5x - 2y = 10 Substituting x = 0 in the given equation, We get; 5 × 0 - 2y = 10 - 2y = 10 - y $=\frac{10}{2}$ y = -5 Thus x = 0 and y = -5 is the solution of 5x - 2y = 10 Substituting y = 0 in the given equation, we get 5x - 2 × 0 = 10 5x = 10$\text{x} = \frac{10}{5}$
x = 2 Thus x = 2 and y = 0 is a solution of 5x - 2y = 10

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Question 33 Marks

Give the geometric representations of the following equations:
$a.$ On the number line.
$b.$ On the Cartesain plane.
$2x + 9 = 0$

Answer

$2x + 9 = 0, 2x = -9,\text{X}=\text{x}=\frac{-9}{2}=-4.5$ Point $A$ represents $-4.5$ on the number line.
On Cartesian plane, equation represents all points on $y$ axis for which $x = -4.5$

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Question 43 Marks

Give the geometrical representation of $2x + 13 = 0$ as an equation in: One variable.

Answer

One variable representation of $2x + 13 = 0$
$2x = -13$
$\text{x}=\frac{-13}{2}=-6\frac{1}{2}$
Point A represents $-\frac{13}{2}.$

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Question 53 Marks

Write two solutions of the form $x = 0, y = a$ and $x = b, y = 0$ for the following equations: $-4x + 3y = 12$

Answer

We are given, $-4x + 3y = 12$
Substituting $x = 0$ in the given equation,
we get; $-4 × 0 + 3y = 12\ 3y = 12 y = 4$
Thus $x = 0$ and $y = 4$ is a solution of the $-4x + 3y = 12$
Substituting $y = 0$ in the given equation,
we get; $-4x + 3 \times 0 = 12 - 4x = 12$ $\text{x} = -\frac{12}{4}$ $x = -3$
Thus $x = -3$ and $y = 0$ is a solution of $-4x + 3y = 12$.

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Question 63 Marks

Give the geometric representations of the following equations:
$a.$ On the number line.
$b.$ On the Cartesain plane.
$x = 2$

Answer

$x = 2$ Point $A$ represents $x = 2$ number line.
On Cartesian plane, eqution represents all points on $y$ axis for which $x = 2$

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Question 73 Marks

Write two solutions for the following equations: $\frac{2}{3}\text{x} - \text{y} = 4$

Answer

We are given,$\frac{2}{3}\text{x} - \text{y} = 4$
Substituting $x= 0$ in the given equation, we get
$\frac{2}{3}\text{x} - \text{y} = 4$
$0 - y = 4$
$y = -4$
Thus $x = 0$ and $y = -4$ is the solution of $\frac{2}{3}\text{x} - \text{y} = 4$
Substituting $x = 6$ in the given equation, we get
$\frac{2}{3}\text{x} - \text{y} = 4$
$-y = 4 - 2$
$y = -2$
Thus $x = 3$ and $y = -2$ is the solution of $\frac{2}{3}\text{x} - \text{y} = 4$

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Question 83 Marks

Write two solutions of the form $x = 0, y = a$ and $x = b, y = 0$ for the following equations: $2x + 3y = 24$

Answer

We are given, $2x + 3y = 24$
Substituting $x = 0$ in the given equation,
we get; $2 \times 0 + 3y = 24 3y = 24$
$\text{y}=\frac{24}{3}$ $y = 8$ Thus $x = 0$ and $y = 8$ is a solution of $2x + 3y = 24$
Substituting $y = 0$ in the given equation, we get; $2x + 3 × 0 = 24\ 2x = 24$
$\text{x} = \frac{24}{2}$ $x = 12$
Thus $x = 12$ and $y = 0$ is a solution of $2x + 3y = 24$

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Question 93 Marks

Give the geometric representations of the following equations:
$a.$ On the number line.
$b.$ On the Cartesain plane.
$y = 3$

Answer

$y = 3$ Point $A$ represents $3$ on number line.
On Cartesian plane, equation represents all points on $x$ axis for which $y = 3$

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Question 103 Marks

If $x=1$ and $y=6$ is a solution of the equation $8 x-a y+a^2=0$, find the values of $a$.

Answer

We are given,
$8 x-a y+a^2=0(1,6)$ is a solution of equation $8 x-a y+a^2=0$
Substituting $x=1$ and $y=6$ in $8 x-a y+a^2=0$, we get $8 \times 1-a \times 6+a^2=0$
$\Rightarrow a^2-6 a+8=0$
Using quadratic factorization $a^2-4 a-2 a+8=0 a(a-4)-2(a-4)=0(a-2)(a-4)=0$
$a=2,4$

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Question 113 Marks

Write two solutions for the following equations: $x = 6y$

Answer

We are given, $x = 6y$ Substituting $x = 0$ in the given equation,
we get $0 = 6y$ $\text{y}=\frac{0}{6}$ $y = 0$ Thus $x = 0$ and $y = 0$ is the solution of $x = 6y$ Substituting $x = 6$ in the given equation,
we get $6 = 6y$ $\text{y}=\frac{6}{6}$ $y = 1$ Thus $x = 6$ and $y = 1$ is the solution of $x = 6y$.

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Question 123 Marks

Write two solutions for the following equations: $\text{x}+\pi\text{y} = 4 $

Answer

We are given, $\text{x}+\pi\text{y} = 4 $
Substituting $x = 0$ in the given equation,
we get $\text{x}+\pi\text{y} = 4 $ $\pi\text{y}=4-0$
$\text{y}=\frac{4}{\pi}$Thus $x = 0$ and $\text{y}=\frac{4}{\pi}$ is the solution of $\text{x}+\pi\text{y} = 4 $
Substituting $x = 6$ in the given equation,
we get $\text{x}+\pi\text{y} = 4 $
$\pi\text{y}=4-0$
$\text{y}=\frac{0}{\pi}$ $\text{y}=0$
Thus $x = 4$ and $y = 0$ is the solution of $\text{x}+\pi\text{y} = 4 $

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Question 133 Marks

Give the geometrical representation of $2x + 13 = 0$ as an equation in: Two variables.

Answer

Two variable representation of $2x + 13 = 0$
$2x + 0y + 13 = 0$
$2x = 13 = 0$
$2x = -13$
$\text{x}=-\frac{13}{2}=-6\frac{1}{2}$
On Cartesian plane, equation represents all points on $y$ axis for which $x = -6.5$

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Question 143 Marks

Find the value of $k$ for which the point $(1, -2)$ lies on the graph of the linear equation $x - 2y + k = 0$.

Answer

We are given $(1, -2)$ lies on the graph of linear equation $x - 2y + k = 0.$
So, the given co-ordinates are the solution of the equation $x - 2y + k = 0.$
Therefore, we can calculate the value of k by substituting the value of given co-ordinates in equation $x - 2y + k = 0.$
Substituting $x = 1$ and $y = -2$ in equation $2x - 3y + 8 = 0$, we get
$1 - 2(-2) + k = 0$
$1 + 4 + k = 0$
$k = -5$

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Question 153 Marks

Write two solutions for the following equations: $3x - 4y = 7$

Answer

We are given, $3x 1 + 4y = 7$
Substituting $x = 1$ in the given equation,
we get $3x\ 1 + 4y = 7 4y = 7 - 3$ $\text{y}=\frac{4}{4}$
Thus $x = 1$ and $y = 1$ is the solution of $3x + 4y = 7$
Substituting $x = 2$ in the given equation,
we get $3 \times 2 + 4y = 7$ $\text{y}=\frac{1}{4}$
$\text{y}=\frac{1}{4}$
Thus $x = 2$ and $\text{y}=\frac{1}{4}$ is the solution of $3x + 4y = 7$

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Question 163 Marks

Give the geometric representations of the following equations:
$a.$ On the number line.
$b.$ On the Cartesain plane.
$y + 3 = 0$

Answer

$y + 3 = 0, y = -3$ Point $A$ represents $-3$ on number line.
On Cartesian plane, equation represents all points on $x$ axis for which $y = -3$.

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Question 173 Marks

If $x = 2a + 1$ and $y = a -1$ is a solution of the equation $2x - 3y + 5 = 0$, find the value of a.

Answer

We are given, $2x - 3y + 5 = 0 (2a + 1, a - 1)$ is the solution of equation $2x - 3y + 5 = 0$.
Substituting $x = 2a + 1$ and $y = a - 1$ in $2x - 3y + 5 = 0$,
We get $2 \times 2a + (1- 3) \times a - 1 + 5 = 0 $
$\Rightarrow 4a + 2 - 3a + 3 + 5 = 0 $
$\Rightarrow a + 10 = 0 $
$\Rightarrow a = -10$

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Question 183 Marks

If the point $(a, 2)$ lies on the graph of the linear equatio $2x - 3y + 8 = 0$, find the value of a.

Answer

We are given $(a, 2)$ lies on the graph of linear equation $2x - 3y + 8 = 0$.
So, the given co-ordinates are the solution of the equation $2x - 3y + 8 = 0$.
Therefore, we can calculate the value of a by substituting the value of given co-ordinates in equation $2x - 3y + 8 = 0$.
Substituting $x = a$ and $y = 2$ in equation $2x - 3y + 8 = 0$, we get
$2 \times a - 3 \times 2 + 8 = 0$
$2a - 6 + 8 = 0$
$2a + 2 = 0$
$2a = -2$
$\text{a}=-\frac{2}{2}$
$a = -1$

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Question 193 Marks

Give the geometric representations of the following equations:
$a.$ On the number line.
$b.$ On the Cartesain plane.
$3x - 5 = 0$

Answer

$3x - 5 = 0 ,3x = 5,\text{x}=\frac{5}{3}=1\frac{2}{3}=1.6($Approx$)$ Point $A$ represents $1\frac{1}{2}$ or $\frac{5}{3}$ on number line.
On Cartesian plane, equation represents all points on $y$ axis for which $x = 1.6$

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