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Linear Equations in Two Variables

Short Answer Type

31. 5. If x = –2, y = 6 is solution of equation 3ax + 2by = 6, then find the value of b from 2(a – 1) + 2(3b – 4) = 4.

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32. The coordinates of the points given in the following table represent some of the solutions of the equation $y equals 3 over 2 space space x space minus 1$

Find the missing values. Also find the coordinates of the points where the line cuts x-axis and y-axis.

$straight y equals 3 over 2 straight x minus 1$
When x = 2, $straight y equals 3 over 2 left parenthesis 2 right parenthesis minus 1 equals 3 minus 1 equals 2$

When y = -4, -4 = $3 over 2 x minus 1$

$rightwards double arrow space space space space space space space space space space space 3 over 2 straight x equals negative 3 rightwards double arrow space space space space space space space space space space space space straight x space equals space minus 2$

When y = 8, 8 = $3 over 2 x equals negative 1$
$rightwards double arrow space space space space space space 3 over 2 straight x equals 9$
$rightwards double arrow$ x = 6

When x = -10, y = $3 over 2 left parenthesis negative 10 right parenthesis minus 1 equals negative 16$

When x = 1, y = $3 over 2 left parenthesis 1 right parenthesis minus 1 equals 1 half$

When y = $negative 5 over 2 comma space space minus 5 over 2 equals 3 over 2 straight x minus 1$

$rightwards double arrow space space space space space space space space space space 3 over 2 straight x equals negative 3 over 2$

$rightwards double arrow$ x = -1

Hence, the completed table is as follows:

$fraction numerator straight x space space space space space space 2 space space space space minus 2 space space space space space space 6 space space space space space minus 10 space space space space space space space 1 space space space space space space minus 1 over denominator space space straight y space space space space space space 2 space space space space minus 4 space space space space space space space 8 space space space space space minus 16 space space space space space begin display style 1 half end style space space space space minus begin display style 5 over 2 end style space space space space end fraction$

For intersection with x-axis, put y = 0

$therefore space space space space space space space space 0 equals 3 over 2 straight x minus 1 rightwards double arrow space space space space space space space straight x equals 2 over 3$

Hence, the point of intersection is $open parentheses 2 over 3 comma space 0 close parentheses.$

For intersection with y-axis, put x = 0

$therefore$ y = -1
Hence, the intersection is (0, -1)

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33. Find the coordinates of the points where the line representing the equation

straight x over 4 equals 1 minus y over 6

cuts the x-axis and the y-axis.

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34. Determine the point on the graph of the equation 2x + 5y = 20 where x-coordinate is

5 over 2

times its ordinate.

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35.

Find four different solutions of the equation

x + 2y = 6.

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36.

Find two solutions for each of the following equations:

(i) 2x – 3y = 12 (ii) 2x – 5y = 0

(iii) 3y – 4 = 0.

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37.

Find the value of a so that the following equation may have x = 1, y = 1 as a solution:

3x + ay = 6

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38.

Write the equation $straight y square root of 3 equals 8 straight x plus square root of 5$ in the form of ax + by + c =0. Check whether (0, 1) and $left parenthesis square root of 3 comma space 9 right parenthesis$ are the solutions of the equation.