3 Marks Question

Question 13 Marks

How many solution(s) of the equation 2x + 1 = x – 3 are there on the:

Number line
Cartesian plane?

Answer

The number of solution(s) of the equation 2x + 1 = x – 3 which are on the number line is one.

2x + 1 = x - 3 ⇒ 2x - x = -3 - 1 ⇒ x = -4
$\therefore$ x = -4 is the solution of the given eqution.

As in the Cartesian Plane the equation can be written as x + 0y = -4.

And for infinitely many values of y we have infinite values of x.
So the number of solution(s) of the equation 2x + 1 = x – 3 which are on the Cartesian plane are infinitely many solutions.

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Question 23 Marks

Show that the points A(1, 2), B(-1, -16) and C(0, -7) lie on the graph of the linear equation y = 9x - 7.

Answer

For A(1, 2), we have 2 = 9(1) - 7 = 9 - 7 = 2
For B(-1, -16), we have -16 = 9(-1) - 7 = -9 - 7 = -16
For C(0, -7), we have -7 = 9(0) - 7 = 0 - 7 = -7
We see that the line y = 9x - 7 is satisfied by the points A(1, 2), B(-1, -16) and C(0, -7).
Therefore, A(1, 2), B(-1, -16) and C(0, -7) are solutions of the linear equation y = 9x - 7 and therefore, lie on the graph of the linear equation y = 9x - 7.

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Question 33 Marks

Draw the graph of the equation represented by a straight Line which is parallel to the X-axis and at a distance 3 units below it.

Answer

The graph of the equation y = -3 is a line parallel to the x-axis and at a distance 3 units below it. So, graph of the equation y = -3 is a line parallel to x-axis and passing through the point (0, -3) as shown in the figure given below:

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Question 43 Marks

If the point (3, 4) lies on the graph of 3y = ax + 7, then find the value of a.

Answer

Since, the point (x = 3, y = 4) lies on the equation 3y = ax + 7, then the equation will be , satisfied by the point.
Now, put x = 3 and y = 4 in given equation, we get
3(4) = a(3) + 7
⇒ 12 = 3a + 7
⇒ 3a = 12 – 7
⇒ 3a = 5
Hence, the value of a is $\frac{5}{3}$.

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Question 53 Marks

If the temperature of a liquid can be measured in kelvin units as x°K or in fahrenheit units as y°F, the relation between the two systems of measurement of temperature is given by the linear equation.
$\text{y}=\frac{9}{5}(\text{x}-273)+32$

find the temperature of the liquid in fahrenheit, if the temperature of the liquid is 313K.
If the temperature is 158°F, then find the temperature in kelvin.

Answer

The linear equation that converts kelvin (x) to Fahrenheit (y) is given by the relation $\text{y}=\frac{9}{5}(\text{x}-273)+32$
When the temperature of the liquid is x = 313°k
$\text{y}=\frac{9}{5}(313-273)+32=\frac{9}{5}\times40+32=72^\circ+32^\circ=104^\circ\text{F}$
When the temperature of the liquid is x = 158°F
$158=\frac{9}{5}(\text{x}-273)+32\Rightarrow\frac{9}{5}(\text{x}-273)=158-32$
$\Rightarrow\text{x}-273=126\times\frac{5}{9}=70$
$\Rightarrow\text{x}-273=70=273+70=343\text{k}$

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Question 63 Marks

Determine the point on the graph of the linear equation 2x + 5y = 19 whose ordinate is $1\frac{1}{2}$ times its abscissa.

Answer

Let x be the abscissa of the given line 2x + 5y = 19, then by given condition, Ordinate $\text{(y)}=1\frac{1}{2}\times\text{Abscissa}$
$\Rightarrow\text{y}=\frac{3}{2}\text{x}\ ....(\text{i})$
On putting $\text{y}=\frac{3}{2}\text{x}$ in given equation, we get
$2\text{x}+5\Big(\frac{3}{2}\Big)\text{x}=19$
$\Rightarrow4\text{x}+15\text{x}=19\times2$
$\Rightarrow4\text{x}+15\text{x}=38$
$\Rightarrow 19\text{x}=38$
$\Rightarrow\text{x}=\frac{38}{19}$
$\therefore\text{x}=2$
On substituting the value of x in Eq. (i) we get
$\text{y}=\frac{3}{2}\times2=3$
$\Rightarrow\text{y}=3$
Heanse, the required point is (2, 3).

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