3. Pair of Linear Equations in Two Variables

*नोट: यह सेक्शन रिवीजन के लिए है। अपना ज्ञान परखने के लिए ऊपर दिए गए क्विज़ में भाग लें।

1. What is the general mathematical form of a linear equation in two variables?
   A) ax + by + c = 0  |  B) ax^2 + bx + c = 0  |  C) ax + b = 0  |  D) ax^3 + bx^2 = 0
2. The geometrical graph of a linear equation in two variables is always exactly a:
   A) Circle  |  B) Straight line  |  C) Parabola  |  D) Ellipse
3. If the graph of a pair of linear equations represents two intersecting lines, the system has exactly:
   A) No solution  |  B) 2 solutions  |  C) 1 unique solution  |  D) Infinitely many solutions
4. What is the mathematical condition for a pair of linear equations to have exactly 1 unique solution?
   A) a1/a2 = b1/b2  |  B) a1/a2 ≠ b1/b2  |  C) a1/a2 = b1/b2 = c1/c2  |  D) a1/a2 = c1/c2
5. If a pair of linear equations gives strictly parallel lines on a graph, the total number of solutions is:
   A) 1  |  B) 2  |  C) 0  |  D) Infinite
6. A given system of linear equations that has at least 1 valid solution is mathematically called:
   A) Consistent  |  B) Inconsistent  |  C) Dependent  |  D) Quadratic
7. Find the exact value of x that simultaneously satisfies the equations x + y = 5 and x - y = 1.
   A) 1  |  B) 2  |  C) 3  |  D) 4
8. If x = 2 and y = 1 is a verified solution of the equation 2x + 3y = k, find the exact value of k.
   A) 5  |  B) 6  |  C) 7  |  D) 8
9. What is the specific condition for a system of linear equations to be dependent and have infinitely many solutions?
   A) a1/a2 ≠ b1/b2  |  B) a1/a2 = b1/b2 ≠ c1/c2  |  C) a1/a2 = b1/b2 = c1/c2  |  D) b1/b2 ≠ c1/c2
10. Identify the exact coordinate point that represents the solution to the equations x = 5 and y = -2.
    A) (-2, 5)  |  B) (5, 2)  |  C) (-5, -2)  |  D) (5, -2)
11. Find the exact value of k if the equations 2x + ky = 1 and 3x - 5y = 7 have a unique intersecting solution.
    A) k = 10/3  |  B) k ≠ -10/3  |  C) k = -10/3  |  D) k ≠ 10/3
12. The pair of linear equations y = 0 and y = -7 graphically has exactly:
    A) 1 solution  |  B) 2 solutions  |  C) Infinitely many solutions  |  D) 0 solutions
13. If x = a and y = b represents the mathematical solution of the equations x - y = 2 and x + y = 4, then find the values of a and b respectively.
    A) 3 and 5  |  B) 5 and 3  |  C) 3 and 1  |  D) -1 and -3
14. For what exact numerical value of k will the equations kx + 3y = k - 3 and 12x + ky = k yield infinitely many continuous solutions?
    A) 3  |  B) -3  |  C) 6  |  D) -6
15. Algebraically determine if the given pair of equations 3x + 2y = 5 and 2x - 3y = 7 is consistent or inconsistent.
    A) Inconsistent  |  B) Consistent  |  C) Dependent  |  D) Invalid equations
16. A fraction mathematically becomes 1/3 when exactly 1 is subtracted from its numerator. What linear equation correctly represents this if the fraction is x/y?
    A) 3x - y = 3  |  B) x - 3y = 1  |  C) 3x + y = 3  |  D) x + 3y = 1
17. The sum of two distinct positive numbers is exactly 35 and their positive difference is exactly 13. Find the two numbers.
    A) 20 and 15  |  B) 26 and 9  |  C) 24 and 11  |  D) 22 and 13
18. The geometrical graph of the simple equation x = 4 is a straight continuous line that is:
    A) Parallel to the x-axis  |  B) Parallel to the y-axis  |  C) Passing through the origin  |  D) Intersecting both axes
19. Find the exact numerical value of y by completely solving the system 2x + 3y = 11 and 2x - 4y = -24.
    A) 2  |  B) -2  |  C) 5  |  D) -5
20. Determine the graphical nature of the two linear lines represented exactly by 3x + 4y = 5 and 6x + 8y = 10.
    A) Intersecting lines  |  B) Parallel lines  |  C) Perpendicular lines  |  D) Coincident lines
21. A motorboat successfully covers 32 km upstream and 36 km downstream in exactly 7 hours. In exactly 9 hours, it covers 40 km upstream and 48 km downstream. Find the boat's speed in completely still water.
    A) 8 km/h  |  B) 10 km/h  |  C) 12 km/h  |  D) 14 km/h
22. Solve the algebraic equations 1/x + 1/y = 5 and 2/x - 3/y = -5 to find the exact numerical value of x.
    A) 1/2  |  B) 1/3  |  C) 2  |  D) 3
23. The 4 angles of a cyclic quadrilateral are A=(4y+20), B=(3y-5), C=(4x), and D=(7x+5). Mathematically determine the exact degree measure of angle A.
    A) 70 degrees  |  B) 110 degrees  |  C) 120 degrees  |  D) 60 degrees
24. For what precise value of k does the mathematical system 3x + y = 1 and (2k-1)x + (k-1)y = 2k+1 structurally have absolutely no solution?
    A) 1  |  B) 2  |  C) 3  |  D) 4
25. If 8 men and 12 boys can successfully finish a piece of work in exactly 10 days while 6 men and 8 boys can finish it in exactly 14 days, find the total time mathematically taken by 1 man working completely alone.
    A) 100 days  |  B) 120 days  |  C) 140 days  |  D) 160 days
26. A distinct two-digit numerical figure is mathematically equal to exactly 4 times the algebraic sum of its individual digits and twice the algebraic product of its digits. Identify the exact original number.
    A) 24  |  B) 36  |  C) 48  |  D) 12
27. Calculate the exact numerical area of the triangle graphically formed by the straight lines x = 0, y = 0, and 2x + 3y = 6.
    A) 2 square units  |  B) 3 square units  |  C) 4 square units  |  D) 6 square units
28. If we are given the complex system 29x + 37y = 103 and 37x + 29y = 95, quickly find the exact value of the expression x + y.
    A) 1  |  B) 2  |  C) 3  |  D) 4
29. A father's current age is exactly 3 times the sum of the ages of his 2 children. After exactly 5 years, his age will be twice the sum. What is the father's current age?
    A) 40  |  B) 45  |  C) 50  |  D) 55
30. Determine the exact critical value of c for which the given system of linear equations cx - y = 2 and 6x - 2y = 3 mathematically has 0 possible solutions.
    A) 3  |  B) -3  |  C) 12  |  D) -12