[5 marks sum]

Question 15 Marks

Draw the graph of the lines $y = x + 2, y = 2x - 1$ and $y = 2$ from $x = -3$ to $4$, on the same graph paper. Check whether the lines drawn are parallel to each other.

Answer

For,
$y = x + 2$
When $x = 0, y = 0 + 2 = 2$
When $x = 5, y = 5 + 2 = 7$
When$ x = -3, y = -3 + 2 = -1$

$x$	$0$	$5$	$-3$
$y$	$2$	$7$	$-1$

For,
$y = 2x - 1$
When $x = 0, y = 2(0) -1 = -1$
When $x = -2, y = 2(-2) -1 = -5$
When $x = 3, y = 2(3) -1 = 5$

$x$	$0$	$-2$	$3$
$y$	$-1$	$-5$	$5$

For,
$y = 2$
This line is parallel to the $x-$axis and passes through $(0, 2)$

The lines are not parallel to each other.

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Question 25 Marks

Draw a graph of the equation $5x - 3y = 1$. From the graph find the value of:$(i) x$, when $y = 8;(ii) y$, when $x = 2$

Answer

We have
$5 x-3 y=1$
$ \Rightarrow-3 y=1-5 x$
$ \Rightarrow 3 y=5 x-1$
$ \Rightarrow y=\frac{5 x-1}{3}$
When $x=-2$
$\Rightarrow y =-\frac{11}{3}=-3.66$
When $x=0$
$\Rightarrow y =-\frac{1}{3}=-0.33$
When $x=2$
$\Rightarrow y=\frac{9}{3}=3$

$x$	$-2$	$-1$	$0$	$1$	$2$
$y$	$-3.66$	$-2$	$-0.33$	$1.33$	$3$

Thus ordered pairs of $5 x -3 y =1$ are $\{(-2,-3.66),(-1,-2),(0,-0.33),(1,1.33),(2,3)\}$.
Hence graph is a below.

$(i) x$, when $y = 8$
From graph we find that $x = 5$, when $y = 8$
$(ii) y$, when $x = 2$
From graph we find that $y = 3$, when $x = 2.$

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Question 35 Marks

Draw a graph of the equation $2x + 3y + 5 = 0$, from the graph find the value of$:\ (i) x$, when $y = -3;(ii) y,$ when $x = 8$

Answer

We have
$2 x+3 y+5=0$
$ \Rightarrow 2 x+3 y=-5$
$ \Rightarrow 3 y=-5-2 x$
$ \Rightarrow y=\frac{-2 x-5}{3}$
When $x =-2$
$\Rightarrow y=-\frac{1}{3}=0.33$
When $x =0$
$\Rightarrow y=-\frac{5}{3}=-1.66$
When $x=2$
$\Rightarrow y=-\frac{9}{3}=-3$

$x$	$-2$	$-1$	$0$	$1$	$2$
$y$	$-0.33$	$-1$	$-1.66$	$-2.33$	$-3$

Thus ordered pairs of $2 x+3 y+5=0$ are $\{(-2,-0.33),(-1,-1),(0,-1.66),(1,-2.33),(2,-3)\}$.
Hence graph is a below.

$(i) x$, when $y = -3$
From graph we find that $x = 2$, when $y = -3$
$(ii) y$, when $x = 8$
From graph we find that $y = -7$, when $ x = 8.$

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Question 45 Marks

Draw a graph for each of the following equations and find the coordinates of the points where the line drawn meets the $x-$axis and $y-$axis$: \frac{2 x}{5}+\frac{y}{2}=1$

Answer

$\frac{2 x}{5}+\frac{y}{2}=1$
$ \Rightarrow \frac{y}{2}=1-\frac{2 x}{5}$
$ \Rightarrow \frac{y}{2}=\frac{5-2 x}{5}$
$ \Rightarrow y=\frac{10-4 x}{5}$
When $x=0, y=\frac{10-4(0)}{5}=2$
When $x=5, y=\frac{10-4(5)}{5}=-2$
When $x =\frac{5}{2}, y=\frac{10-4\left(\frac{5}{2}\right)}{5}=0$

$x$	$0$	$5$	$\frac{5}{2}$
$y$	$2$	$-2$	$0$

Plotting the points $(0,2),(5,-2)$ and $\left(\frac{5}{2}, 0\right)$, we get a line segment as shown in the figure.

The line meets the $x-$axis at $\left(\frac{5}{2}, 0\right)$ and $y-$axis at $(0,2)$.

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Question 55 Marks

Draw a graph of the equation $2x - 3y = 15$. From the graph find the value of:$(i) x$, when $y = 3;(ii) y$, when $x = 0$

Answer

We have
$2 x-3 y=15$
$ \Rightarrow-3 y=15-2 x$
$ \Rightarrow 3 y=2 x-15$
$ \Rightarrow y=\frac{2 x-15}{3}$
When $x =-2$
$\Rightarrow y =-\frac{19}{3}$
$ =-6.34$
When $x =0$
$\Rightarrow y=-\frac{15}{3}=-5$
When $x =2$
$\Rightarrow y=-\frac{11}{3}=-3.66$

$x$	$-2$	$-1$	$0$	$1$	$2$
$y$	$-6.34$	$-5.66$	$-5$	$-4.34$	$-3.66$

Thus ordered pairs of $2 x-3 y=15$ are $\{(-2,-6.34),(-1,-5.66),(0,-5),(1,-4.34),(2,-3.66)\}$.
Hence graph is a below.

$(i) x$, when $y = 3$
From graph we find that $x = 12$, when $y = 3$
$(ii) y$, when $x = 0$
Fro graph we find that $y = -5$, when $x = 0.$

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Question 65 Marks

Draw a graph of the equation $3x - y = 7$. From the graph find the value of$:\ (i) y$, when $x = 1;(ii) x$, when $y = 8$

Answer

We have
$3x - y = 7$
$\Rightarrow -y = 7 - 3x$
$\Rightarrow y = 3x - 7$
When $x = -2$
$\Rightarrow y = -6 - 7$
$= -13$
When $x = 0$
$\Rightarrow y = -7$
When $x = 2$
$\Rightarrow y = 6 - 7$
$= -1$

$x$	$-2$	$-1$	$0$	$1$	$2$
$y$	$-13$	$-10$	$-7$	$-7$	$-1$

Thus ordered pairs of $3x - y = 7$ are ${(-2, -13), (-1, -10), (0, -7), (1, -4), (2, -1)}$.
Hence graph is as below.
$(i) y,$ when $x = 1$
From graph we find that $y = -4$, when$ x = 1$
$(ii) x,$ when $y = 8$
From graph we find that $x = 5,$ when $y = 8.$

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Question 75 Marks

Draw a graph of each of the following equations$:\frac{x-2}{3}-\frac{y+1}{2}=0$

Answer

$\frac{x-2}{3}-\frac{y+1}{2}=0$
$ \Rightarrow \frac{x-2}{3}-\frac{y+1}{2}$
$ \Rightarrow 2(x-2)=3(y+1)$
$ =2 x-4=3 y+3$
$ \Rightarrow 3 y=2 x-7$
$ \Rightarrow y=\frac{2 x-7}{3}$
When $x=2, y=\frac{2(2)-7}{3}=-1$
When $x=-1, y=\frac{2(-1)-7}{3}=-3$
When $x=-2.5, y=\frac{2(-2.5)-7}{3}=4$

$x$	$2$	$-1$	$-2.5$
$y$	$-1$	$-3$	$-4$

Plotting the points $(2, -1), (-1, -3)$ and $(-2.5, -4)$, we get a line $AB$ as shown in the figure.

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Question 85 Marks

Draw the graph of the lines represented by the equations $5y = 3x + 1$ and $y = 2x + 3$ on the same graph. Find the coordinates of the point where they intersect.

Answer

For equation $(1),$
$5 y=3 x+1$
$ y=\frac{3 x+1}{5}$
When $x =-2$,
$y=\frac{3(-2)+1}{5}=-1$
When $x =8$,
$y=\frac{3(8)+1}{5}=5$
When $x =-7$,
$y=\frac{3(-7)+1}{5}=-4$

$x$	$-2$	$8$	$-7$
$y$	$-1$	$5$	$-4$

For equation $(2),$
$y=2 x+3$
When $x=0, y=2(0)+3=3$
When $x=-1, y=2(-1)+3=1$
When $x=-7, y=2(-7)+3=-11$

$x$	$0$	$-1$	$-7$
$y$	$3$	$1$	$-11$

$(-2, -1)$

$\therefore$ The point of intersection of the $2$ lines is $(-2, -1).$

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Question 95 Marks

Draw the graph of the lines represented by the equations $2x - y = 8$ and $4x + 3y = 6$ on the same graph. Find the co$-$ordinates of the point where they intersect.

Answer

We have
$2 x-y=8$
$ \Rightarrow-y=8-2 x$
$ \Rightarrow y=2 x-8$
When
$x =-2$
$ \Rightarrow y =-4-8$
$ =-12$
When
$x=0$
$ \Rightarrow y=-8$
When
$x=2$
$ \Rightarrow y=4-8$
$ =-4$

$x$	$-2$	$-1$	$0$	$1$	$2$
$y$	$-12$	$-10$	$-8$	$-6$	$-4$

Thus ordered pairs of $2 x-y=8$ are $\{(-2,-12),(-1,-10),(0,-8),(1,-6),(2,-4)\}$.
Also,
$4 x+3 y=6$
$ \Rightarrow 3 y=6-4 x$
$ \Rightarrow y=\frac{6-4 x}{3}$
When $x =-2$
$\Rightarrow y =\frac{6+8}{3}=4.66$
When $x =0$
$\Rightarrow y =\frac{6}{3}=2$
When $x=2$
$\Rightarrow y=\frac{6-8}{3}=-0.66$

$x$	$-2$	$-1$	$0$	$1$	$2$
$y$	$4.66$	$3.33$	$2$	$0.66$	$-0.66$

Thus ordered pairs of $4x + 3y = 6$ are $\{(-2, 4.66), (-1, 3.33), (0.2), (1, 0.66), (2, -0.66)\}.$

The point of intersection is $(3, -2).$

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Question 105 Marks

Draw the graph of the lines represented by the equations $3x - 2y = 4$ and $x + y = 3$ on the same graph. Find the coordinates of the point where they intersect. State, whether the lines are perpendicular to each other.

Answer

We have
$3 x -2 y =4$
$ \Rightarrow-2 y =4-3 x$
$ \Rightarrow 2 y =3 x -4$
$ \Rightarrow y =\frac{3 x -4}{2}$
When $x=-2$
$\Rightarrow y=\frac{-6-4}{2}=-5$
When $x=0$
$\Rightarrow y=-\frac{4}{2}=-2$
When $x=2$
$\Rightarrow y=\frac{6-4}{2}=1$

$x$	$-2$	$-1$	$0$	$1$	$2$
$y$	$-5$	$-3.5$	$-2$	$-0.5$	$1$

Thus ordered pairs of $3 x-2 y=4$ are $\{(-2,-5),(-1,-3.5),(0,-2),(1,-0.5),(2,1)\}$.
Also,
$x+y=3$
$ \Rightarrow y=3-x$
When $x=-2$
$\Rightarrow y =4+2$
$ =6$
When $x =0$
$\Rightarrow y =3$
When $x =2$
$\Rightarrow y=4-2$
$ =2$

$x$	$-2$	$-1$	$0$	$1$	$2$
$y$	$5$	$4$	$3$	$2$	$1$

Thus ordered pairs of $x + y = 3$ are $\{(-2, 5), (-1, 4), (0, 3), (1, 2), (2, 1)\}.$

The point of intersection is $(2, 1).$

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Question 115 Marks

Draw the graph of the lines represented by the equations $x + y = 4$ and $2x - y = 2$ on the same graph. Find the coordinates of the point where they intersect

Answer

For,
$x + y = 4$
$y = 4 - x$
When $x = 3, y = 4 - 3 = 1$
When $x = 0, y = 4 - 0 = 4$
When$ x = -1, y = 4 - (-1) = 5$

$x$	$3$	$0$	$-1$
$y$	$1$	$4$	$5$

For,
$2x - y = 2$
$y = 2x - 2$
When $x = 3, y = 2(3) - 2 = 4$
When $x = 0, y = 2(0) -2 = -2$
When $x = -1, y = 2(-1) -2 = -4$

$x$	$3$	$0$	$-1$
$y$	$4$	$-2$	$-4$

Plotting thses co$-$ordinates on the graph, we get the lines shown as$:\ (2, 2)$

The point of intersection is $(2, 2).$

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Question 125 Marks

Draw the graph of a line $2x + 3y = 6$. From the graph, find the slope and $y-$intercept of the line.

Answer

$2 x+3 y=6$
$ 3 y=6-2 x$
$ y=\frac{6}{3}-\frac{2}{3} x$
$ \therefore y=2-\frac{2}{3} x$
When $x=3, y=2-\frac{2}{3}(3)=0$
When $x=-3, y=2-\frac{2}{3}(-3)=4$
When $x=0, y=2-\frac{2}{3}(0)=2$

$x$	$3$	$-3$	$0$
$y$	$0$	$4$	$2$

Plotting the points $(3,0),(-3,4)$ and $(0,2)$, we get line segment as shown in the figure.
$\therefore$ The $y$-intercept is $2 .$
Slope
$=\frac{\text { Change in } y}{\text { Change in } x}$
$ =\frac{2-0}{0-3}$
$ =\frac{2}{-3}$
$ \therefore$ Slope
$ =\frac{-2}{3} .$

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MATHEMATICS [5 marks sum]

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