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Important Questions for Class 12 Maths Linear Equations in Two Variables have been provided in the article. A linear equation in two variables is an algebraic equation, that is illustrated as ax + by + c = 0 (wherein, a, b, and c are basically real numbers or constants). The equation helps us determine the variables. A linear equation has the highest variable power of 1. As opposed to it, a non–linear equation has the power of 2 or over. It is otherwise also referred to as a 1-degree equation.
Very Short Answer Questions [1 Mark Question]
Ques. Consider that (1, -2) is a dedicated answer to the linear equation 2x – y = p. Analyzing the question, determine the value of p.
Ans. As per linear equation in two variables, the following equation can be mentioned
= 2x - y = p
After substituting with the respective values, it can be said,
x=1 and y=-2.
2 (1)- (-2) =p
2+2 =p
Hence, it can be further said that, p=4
Ques. Show x/4 – 3y = – 7 in the form of ax + by + c = 0.
Ans. As per the given equation, x/4 – 3y = – 7
Henceforth, it can be said, x-12y+28 = 0
Which is also simultaneously in the form of ax+ by+ c=0
Ques. Find the value of k. If x = 0, y = 8 is a solution of 3x – 6y =k.
Ans. Since we know x = 0 and y = 8,
Put these values in the given equation, which is -
3x - 6y = k
3 (0) - 6 (8) = k
0 - 48 = k
Or k = - 48
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Ques. Considering that a pen costs Rs. 2, which is half times that of the cost of a pencil. Analyzing the same, highlight the expression as a linear equation in two variables.
Ans. Suppose the cost of a pen is Rs x and the cost of a pencil is rs y.
According to the statement,
X= 2 ½ y
2x =5y
Or, 2x-5y=0
Ques. Express x in term of y : x/7 + 2y = 6
Ans. As per the aforementioned equation, it can be denoted –
x/7 + 2y =6
or x/7 = 6-2y
Thus, x= 7(6-2y).
Ques. In a one-day international cricket match, Suresh Raina and MS Dhoni together scored 198 runs. Now write down the statement as a linear equation in two variables.
Ans. Consider that the runs scored by Raina is x and the runs scored by Dhoni is y.
According to the statement, we have
x + y = 198,
x + y – 198 = 0
Ques. Write down the linear equation representing a line that is parallel to the y-axis and is at a distance of 2 units on the left side of the y-axis. [CBSE]
Ans. Here, the required equation is parallel to the y-axis at a distance of 2 units on the left side of the y-axis.
x = – 2
x + 2 = 0
Short Answer Questions [2 Marks Question]
Ques. As we know, In some countries temperature is measured in Fahrenheit, and on the other hand, in countries like India, it is measured in Celsius. Now, we are giving you a linear equation that converts Fahrenheit to Celsius:
F=[9/5] c+32. If the temperature is – 40°C, determine the temperature in Fahrenheit.
Ans. Given linear equation is
F= [9/5] c+32
Put c = -40°, we have
F = 9/5(-40°) + 32
F=- 72 +32
F = -40°
Ques. What will be the value of k, if (1, -1) is a solution of the equation 3x-ky = 8. Besides that, find the coordinates of another point lying on its graph.
Ans. Since we’re aware (1, -1) is a solution of the equation 3x - ky = 8
So, 3 (1) – k (-1) =8
Or k= 8-3
Or k=5
Thus, the given equation is
3x-5y =8
Put x=6, then
y = \(\frac{3*6-8}{5} \) = \(\frac{18-8}{5} \) = \(\frac{10}{5} \)= 2Thus, the coordinates of another point lying on the graph of 3x -5y =8 is (6,2)
Ques. If (p, 2p + 1) is the solution of the linear equation 4x + 3y = 23. Find the value of p.
Ans. Suppose (p, 2p, +1) is the solution of the linear equation 4x +3y =23. Then we have,
4p + 3 (2p+1) =23
Or, 4p + 3(2p+1) =23
Or, 10p =20
Or, we can say p=2
Ques. Highlight and evaluate the value of m, if (5,8) is a solution of the equation 11 x-2y = 3m, then find one more solution also of this given equation.
Ans. Here, (5,8) is the solution of the equation
11x - 2y = 3m
11(5) - 2(8) = 3m
55 - 16 = 3m
39 = 3m or m =39
Now, equation becomes 11x - 2y = 39
Take x = 3, we have
11(3) - 2y = 39
33 - 2y = 39
-2y = 6
y = -3
Thus, (3, -3) is another solution of the given equation.
Ques. If π x + 3y = 25, write y in terms of x and also, find the two solutions to this equation.
Ans. Given the equation is –
When x=0, then y = 25 /3
When x=1, then y= \(\frac{25-{\pi}}{3} \)Hence the two solutions are x =0, y = 25/3
And x=1, y= 25–π / 3
Ques. Find four solutions of 2x - y = 4.
Ans. In accordance with the given equation, we have 2x - y = 4
y = 2x - 4
put x=0 in (i), we have
y= 2(0) – 4 = -4
Thus, put x=1 in (1), we have
y= 2(1) -4 =-2
Now, after replacing x = 2 in (i), we can obtain y = 2 (2) - 4 = 0
Now, after replacing x = 3 in (i), we can obtain y = 2 (3) - 4 = 2
Henceforth, (0, -4), (1, -2), ( 2, 0), and (3, 2) will be the final answer of the aforementioned equation.
Ques. Give the equation of two lines on the same plane which are intersecting at the point (2, 3).
Ans. Because there are infinite lines passing through the point (2, 3).
Let’s assume the first linear equation is x + y = 5 and the second equation is 2x + 3y = 13.
We can see that the lines represented by both equations intersect at the point (2, 3).
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Long Answer Questions [3 Marks Question]
Ques. Tell the value of a for which the equation 2x + ay = 5 has (1, -1) as a solution. And find out two more solutions for the equation obtained. [CBSE March 2011 ]
Ans. Here, (1, -1) is a solution of the equation
2x + ay = 5
2(1) + a (-1) = 5
2- a= 5
-a = 3
a = -3
After replacing the value of a,
2x - ay = 5, we’re supposed to obtain 2x - 3y = 5
Now, after considering x = 4, we need to consider
2 (4) - 3y = 5
→ - 3y = -3
→ y = 1
Now, consider x = - 2,
2 (-2) - 3y = 5
Thus, - 4 - 3y = 5
y = -3
Therefore, the results show that (4,1 and (-2, -3) are the two solutions
Ques. A fraction becomes 1/4 when 2 is subtracted from the numerator and 3 is added to the denominator. Now, express this situation as a linear equation in two variables. And also, find two solutions for this. [CBSE-15-17DIG1U]
Ans. Suppose the numerator and denominator of the given fraction be respectively x and y. According to the statement, we can obtain
\(\frac{x-2}{y+3}\) = \(\frac{1}{4}\)
= 4x – 8 = y + 3
= 4x – y – 11 = 0
which is the required linear equation,
If y = 1, then x = 3.
But, if y = 5, then the value of x = 4.
Therefore, the solutions are: (3, 1) and (4, 5)
Ques. Draw the graph of the linear equation y = 2/3 x + 1/3. Evaluate from the graph that (7, 5) is a solution to the linear equation.
Ans. 
Ques. The route of an airplane is explained by the equation 3x – 4y = 12. Represent the graph graphically. Also, represent that the point (- 4, – 6) lies on the graph. [CBSE-15-NS72LP7]
Ans. The two given equations are:-
Very Long Answer Questions [5 Marks Question]
Ques. In the following graph paper, draw the straight line 3x – 2y = 4 and x + y – 3 = 0. Also, find and write their point of intersection on the graph.
Ans. The two given equations are
Ques. Assume that (2,3) and (4, 0) lie on the graph of linear equation ax + by = 1. Analyzing the same, determine the value of a and b. Plot the graph of the equation obtained. [CBSE March 2012]
Ans.
Which requires a linear equation.
Put x = 0 in eq. (iii)
3(0) + 2y = 12
y = 6
Put x = 2 in eq. (iii)
3(2) + 2y = 12
y = 3
Put x = 4 in eq. (iii)
3(4) + 2y = 12
2y = 0
y=0
We have the following table :
With the points (0, 6), (2, 3) and (4,0) and then Joining them, we obtained the graph of
3x + 2y = 12.
Ques. Write down the linear equations of the lines drawn in the following graph :
And also calculate the area enclosed between these lines.
Ans.
Equations of the lines drawn in the graph are as follows-
X= -1 or x+1=0
X = 2 or x - 2 = 0
Y = 1 or y - 1 = 0
And y = 3 or y - 3 = 0
It forms a rectangle of dimensions 3 units/2 units. Hence, the area enclosed between the given lines = 6 sq units.
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