ID : in-9-linear-equations-in-two-variables [1] Class 9 Linear Equations in Two Variables For more such worksheets visit www.edugain.com Answer the questions (1) A telecom operator charges Rs. 1.1 for the first minute and Rs. 1 per minute for subsequent minutes of a call. If duration of call is represented as x, and amount charged is represented as y, find the linear equation for this relationship. (2) Find the value of w. (3) Find the equation of straight line which is parallel to x-axis, and is at a distance of d from x-axis is (4) The positive solutions of the equation ux + vy = w always lie in which quadrant?
(5) Find the linear equation represented in the graph below ID : in-9-linear-equations-in-two-variables [2] (6) In the graph of the linear equation 3x + 5y = 14, there is a point such that its ordinate is one third of abscissa. Find coordinates of the point. (7) The Cab fare in Bangaluru is Rs. 21 for the first kilometer and Rs. 14 per kilometer for subsequent distance covered. If distance is represented as d, and fare is represented as f, find the linear equation for this relationship. (8) If point (2, 4) lies on the graph of linear equation 5x + p y = 22, find the value of p. (9) At what point does line represented by the equation 8x + 4y = -4 intersects a line which is parallel to the y-axis, and at a distance 3 units from the origin and in the negative direction of x-axis. Choose correct answer(s) from the given choices (10) If x + 2y = 5 then which of the following values of x and y will hold true? a. x = 3, y = 5 2 b. x = 1, y = 4 2 c. x = 4 3, y = 5 2 d. x = 1, y = 5 2
ID : in-9-linear-equations-in-two-variables [3] (11) If a number is subtracted from both side of a equation, then solution of the equation a. Will also decrease by same number b. May or may not change depending on the equation c. Changes d. Remains the same (12) The graph of equation for the line x = p is a line a. parallel to x-axis at a distance p units from the origin b. parallel to y-axis at a distance p units from the origin c. making an intercept p on the y-axis d. making an intercept p on both the axes (13) If graph of the equation y = mx + c passes through the origin, what is the value of c. a. -1 b. 2 c. 0 d. 1 (14) A point of the form (0, b) lies on the line a. x = 0 b. y = 0 c. x + y = 0 d. x = y (15) Equation 9x + 4y = 10 has: a. Two solutions b. No solution c. Infinitely many solutions d. A unique solution 2017 Edugain (www.edugain.com). All Rights Reserved Many more such worksheets can be generated at www.edugain.com
Answers ID : in-9-linear-equations-in-two-variables [4] (1) y = x + 0.1 We are given the following facts - The charge for the first minute is is Rs. 1.1 - The charge per minute after that is Rs. 1 We can see that the charge will be dependent on the time spent in minutes So we set y on the left hand side y = Some linear function of x We know that after the first minute, the rate is Rs. 1 per minute. So if the call lasts for x minutes, there will be a charge Rs. 1.1 for the first minute, and Rs. 1 for x - 1 minutes. Step 4 This means the equation is y = 1.1 + ((x - 1) x 1) Step 5 Simplifying, we get y = x + 0.1 (2) v/u Equation of line y = mx + c Since it goes through center, c is 0, hence equation is y = m x For first point x = v and y = wv, hence wv = m v m = w (1) For second point x = u and y = v, hence v = m u m = v/u (1) Step 4 On comparing two equations, w = v/u
(3) y = d ID : in-9-linear-equations-in-two-variables [5] If a line is parallel to the x-axis, then y value of it is constant for all values of x. Take a look at the image to see this case Further, if the line is distance d away from the x-axis, it also means that this constant value of y is d. So the equation for that line is y=d (4) First quadrant It is given that the solution of the equation is positive, it means the values of the x and y is positive. Therefore, x>0 and y>0 Since we know that if the values of x and y in the first quadrant is positive, i.e. greater than 0 and hence we can say that the positive solutions of the equation px + qy + r = 0 always lie in first quadrant as shown below.
(5) y = - 2 ID : in-9-linear-equations-in-two-variables [6] The general equation of a line is y=mx+c So we have to find m and c To find c, note from the equation that c is the value of y when x=0 (i.e. the equation becomes y=m*0 + c, or y=c). Look at the graph to see if this is a vertical line. If it is not (we'll see the case where it is later in this tip), then what the value of y is when the equation crosses the vertical axis We see that the value of y at this point is -2. So c=-2 The next part is finding m The best way to consider m is to think of it as the slope of the line. Think of it as the change in y for a given change in x. Consider the two equations, y 1 = mx 1 + c, and y 2 = mx 2 + c Now we subtract the first equation from the second We get y 1 - y 2 = mx 1 + c - (mx 2 + c) Simplifying, (y 1 - y 2 ) = m(x 1 - x 2 ) or m = (y 1 - y 2 )/(x 1 - x 2 ) Now, substitute the two points seen in the graph. m = (-2 - (-2))/(2 - (-2)) Also, note that this is the reason why we don't apply this when the line is vertical, because the denominator would be 0, and the equation is meaningless This is solved to get the value of m, and get the answer m=0 Now, if the line is a vertical one, then you can solve it by inspection. So the answer is y= - 2.
(6) (3, 1) ID : in-9-linear-equations-in-two-variables [7] We are given the following: a. The equation is 3x + 5y = 14 b. The line has a point where the value of the ordinate is one third the value of the abscissa. The second fact implies that the point is of the form (x, x ). 3 Substituting y = x 3, in the equation 3x + 5y = 14 we get: 3x + 5x 3 or, x = 3 = 14 We have x = 3, which means the coordinates of the point will be (3, 1). (7) f = 14d + 7 We are given the following facts - The fare for the first kilometer is Rs. 21 - The fare per kilometer after that is Rs. 14 We can see that the fare will be dependent on the distance So we set f on the left hand side f = Some linear function of d We know that after the first kilometer, the rate is Rs. 14 per kilometer. So if we travel d kilometers, we will get charged Rs. 21 for the first kilometer, and Rs. 14 for d - 1 kilometers. Step 4 This means the equation is f = 21 + ((d - 1) x 14) Step 5 Simplifying, we get f = 14d + 7
(8) 3 ID : in-9-linear-equations-in-two-variables [8] We know the following facts - The equation of the line is 5x + p y = 22 - The point (2,4) lies on the line Substitute x=2 and y=4 in the equation 5 x 2 + p x 4 = 22 Solve this to find that the value of p is 3. (9) (-3, 5) Let's consider the second line first. The line which is parallel to the y-axis and is at a distance 3 units from the origin in the negative direction of the x-axis is defined by the following equation x=-3 So, now we know that at the point of intersection, the value of x = -3 The equation of the first line is 8x + 4y = -4 Subtituting for x with the value -3 in this equation, we get y = -3 So the answer is that the intersection is at the point (-3, 5) (10) b. x = 1, y = 4 2 Here, we are given an equation with 2 variables, i.e., x + 2y = 5. It is not possible to find the value of x and y from the given equation, as we require 2 equations to find the value of 2 variables. However, we can substitute the values of x and y given in the options to check the correctness of the equations. Among the given options, we can see that only x = 1, y = 4 2 satisfies the equation. x + 2y = 5 1 + 2 4 2 = 5 5 = 5 Hence, option b is the correct answer.
(11) d. Remains the same ID : in-9-linear-equations-in-two-variables [9] Think of this in simple terms. If two values (let it be anything - weights, lengths, coins, equations etc.) are equal, and you add or remove some amount from both of them, the resulting values will also be equal. That is the principle here, and the answer is that the solution to the equation will remain the same. (12) b. parallel to y-axis at a distance p units from the origin If the equation for the line is x = p, this implies that the value of x is always p irrespective of the value of y. What this means is that this line is parallel to y-axis at a distance p units from the origin. (13) c. 0 For a line to pass through point (0,0), it's equation need to satisfy for x = 0 and y = 0 Lets substitute these values of x and y in the equation and check, y = mx + c 0 = m 0 + c c = 0 Therefore, the graph of the equation y = mx + c will pass through the origin if value of c is 0 (14) a. x = 0 There are of course, infinite lines that can pass through a given point, but we have to choose from the four possibilities presented. The point specified is ((0, b)). Out of the four options the only one it actually can match is x = 0 (15) c. Infinitely many solutions For linear equations in two variables, we need at least two equations to find a unique solution for the pair. A single linear equation can assume infinitely many values of the variables, which is the case with the question. The equation will have infinitely many solutions.