RowQ
The Vault
RowQ
The Vault
CBSE Class 9 Maths · 11 questions · 26 marks
One equation like 2x + 3y = 12 no longer pins down a single answer — it describes a whole straight line of possible answers. Learning to move fluently between the equation, a table of solutions and the drawn graph is the real skill of this chapter. It also explains why the x-axis is simply the line y = 0.
How many solutions does the linear equation 5x - 2y = 9 have?
Answer
It has infinitely many solutions. For any value you choose for x, the equation gives a matching value of y, and every such pair (x, y) is a solution. Geometrically these pairs are the points of a straight line, and a line contains infinitely many points. A unique solution would require a second independent equation.
Which of the following ordered pairs is a solution of 4x + 3y = 25?
Answer
(4, 3) is a solution. Substituting x = 4 and y = 3 gives 4(4) + 3(3) = 16 + 9 = 25, which matches the right-hand side exactly. The others all fail: (1, 6) gives 4 + 18 = 22, (2, 6) gives 8 + 18 = 26, and (5, 2) gives 20 + 6 = 26.
The graph of the equation y = -3 is a straight line that is:
Answer
It is parallel to the x-axis, 3 units below it. The equation places no restriction on x, so x may take any value while y is fixed at -3. All such points lie at the same depth below the x-axis, forming a horizontal line 3 units below it. If the constant were 0 the line would coincide with the x-axis instead.
If (3, 0) is a solution of the equation 2x + ky = 6, then the value of k is:
Answer
k can be any real number. Substituting x = 3 and y = 0 gives 2(3) + k(0) = 6, that is 6 + 0 = 6, which is true no matter what k equals. Because the y-coordinate is zero, the term containing k vanishes and places no constraint on it.
Assertion (A): The graph of the equation 3x + 5y = 0 passes through the origin. Reason (R): A linear equation of the form ax + by = 0 is always satisfied by the point (0, 0).
Answer
Both A and R are true and R is the correct explanation of A. Substituting x = 0 and y = 0 into ax + by = 0 gives 0 = 0, which is true for all a and b, so R is a correct general statement. Applying it to 3x + 5y = 0 shows that (0, 0) is a solution, so the line does pass through the origin, making A true and directly explained by R.
Write the equation 7x = 4y - 21 in the standard form ax + by + c = 0 and state the values of a, b and c.
Answer
Bring every term to the left-hand side: 7x - 4y + 21 = 0. Comparing with ax + by + c = 0 gives a = 7, b = -4 and c = 21. (Multiplying throughout by -1 gives the equally valid form -7x + 4y - 21 = 0, so the coefficients are determined only up to a common non-zero factor.)
Find four different solutions of the equation 3x - y = 5.
Answer
Rearrange to y = 3x - 5, then choose convenient values of x. If x = 0, y = -5, giving (0, -5). If x = 1, y = -2, giving (1, -2). If x = 2, y = 1, giving (2, 1). If x = 4, y = 7, giving (4, 7). So four solutions are (0, -5), (1, -2), (2, 1) and (4, 7). Infinitely many others exist, since any value of x yields a solution.
If the point (a, 2a) lies on the graph of 5x - 2y = 12, find the value of a.
Answer
Substitute x = a and y = 2a into the equation: 5a - 2(2a) = 12 5a - 4a = 12 a = 12. So a = 12, and the point is (12, 24). Checking: 5(12) - 2(24) = 60 - 48 = 12, as required.
Draw the graph of the equation 2x + y = 8. From your graph, (a) read the coordinates of the point where the line meets the x-axis, (b) read where it meets the y-axis, and (c) find the area of the triangle formed by this line and the two axes.
Answer
Rearrange to y = 8 - 2x and build a table of solutions: x = 0 gives y = 8, so (0, 8). x = 4 gives y = 0, so (4, 0). x = 2 gives y = 4, so (2, 4). Plot these three points and join them; they lie on one straight line, which confirms the plotting is accurate. (a) The line meets the x-axis where y = 0, at the point (4, 0). (b) The line meets the y-axis where x = 0, at the point (0, 8). (c) The line together with the axes encloses a right-angled triangle with vertices (0, 0), (4, 0) and (0, 8). Its base along the x-axis is 4 units and its height along the y-axis is 8 units. Area = (1/2) × base × height = (1/2) × 4 × 8 = 16 square units.
A cycle repair shop charges a fixed inspection fee plus a fixed rate per hour of labour. A repair taking 2 hours costs Rs 340 and one taking 5 hours costs Rs 700. (a) Taking the fixed fee as Rs c and the hourly rate as Rs m, form two linear equations. (b) Find m and c. (c) Write the cost y for x hours as a linear equation in two variables. (d) Find the cost of a 4-hour repair.
Answer
(a) Total cost = fixed fee + rate × hours, so: 2m + c = 340 ...(i) 5m + c = 700 ...(ii) (b) Subtract (i) from (ii): 3m = 360, so m = 120. Substitute into (i): 2(120) + c = 340, so 240 + c = 340 and c = 100. The hourly rate is Rs 120 and the fixed inspection fee is Rs 100. (c) The relation is y = 120x + 100, or in standard form 120x - y + 100 = 0. (d) For x = 4: y = 120(4) + 100 = 480 + 100 = Rs 580.
Read the following and answer the questions that follow: A water tank at a school already holds 30 litres. A tap fills it at a steady 12 litres per minute. Let y litres be the quantity of water in the tank after x minutes of filling. (a) Write a linear equation in two variables connecting x and y. (b) How much water is in the tank after 7 minutes? (c) After how many minutes does the tank contain 210 litres? (d) What does the y-intercept of the graph of this equation represent in this situation?
Answer
(a) The water present equals the starting amount plus the amount added, so y = 12x + 30, which in standard form is 12x - y + 30 = 0. (b) Put x = 7: y = 12(7) + 30 = 84 + 30 = 114 litres. (c) Put y = 210: 210 = 12x + 30, so 12x = 180 and x = 15. The tank holds 210 litres after 15 minutes. (d) The y-intercept is found by putting x = 0, giving y = 30. It represents the 30 litres already in the tank before the tap was opened, that is, the initial quantity of water at time zero.
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