ID : cn-9-linear-equations-in-two-variables [1] Grade 9 Linear Equations in Two Variables For more such worksheets visit www.edugain.com Answer the questions (1) In the graph of the linear equation 4x + 5y = 24, there is a point such that its ordinate is 3 more than its abscissa. Find coordinates of that point. (2) Find the point where linear equation 3x + 2y = 2 intersects with y-axis. (3) At what point does line represented by the equation 2x + 3y = 27 intersects a line which is parallel to the x-axis, and at a distance 5 units from the origin and in the positive direction of y-axis. (4) In the graph of the linear equation 5x + 3y = 11, there is a point such that its ordinate is twice of abscissa. Find coordinates of the point. (5) The positive solutions of the equation ex + fy + g = 0 always lie in which quadrant? (6) Find the equation of straight line which is parallel to x-axis, and is at a distance of d from x-axis is (7) Find the linear equation represented in the graph below (8) If point (4, 5) lies on the graph of linear equation 3x + b y = 22, find the value of b.
ID : cn-9-linear-equations-in-two-variables [2] Choose correct answer(s) from the given choices (9) Equation 4x + 3y = 7 has a unique solution if x and y are a. Rational Numbers b. Real Numbers c. Positive Real Numbers d. Natural Numbers (10) If graph of the equation y = mx + c passes through the origin, what is the value of c. a. 1 b. -1 c. 0 d. 2 (11) A point on line x = y is of the form a. (d, d) b. (d, -d) c. (0, d) d. (d, 0) (12) A point of the form (d, 0) lies on the line a. y = 0 b. x = 0 c. x = y d. x + y = 0 (13) The equation of x-axis is: a. x + y = 0 b. x = 0 c. y = 0 d. x = y (14) Equation 4x + 2y = 5 has: a. Two solutions b. Infinitely many solutions c. No solution d. A unique solution (15) The graph of equation for the line x = b is a line a. parallel to x-axis at a distance b units from the origin b. making an intercept b on both the axes c. parallel to y-axis at a distance b units from the origin d. making an intercept b on the y-axis 2017 Edugain (www.edugain.com). All Rights Reserved Many more such worksheets can be generated at www.edugain.com
Answers ID : cn-9-linear-equations-in-two-variables [3] (1) (1, 4) We are given the following facts: The equation is 4x + 5y = 24 The line has a point where the value of the ordinate is 3 more the value of the abscissa The second fact implies the point is of the form (x,x + 3) Substituting this into the equation, we get 4x + 5(x - 3) = 24 Step 3 Solving for this gets us the value of x = 1. From this we can find y = x + 3 = 4 (2) (0, 1) We are told to find the point where the equation intersects with the y axis. Now, at that point the value of x will be zero. So we need to substitute x=0 into the equation. From there, we can then solve to find the value of y to be 0. So the point is (0,1) (3) (6, 5) Let's consider the second line first. The line which is parallel to the x-axis and is at a distance 5 units from the origin in the positive direction of the y-axis is defined by the following equation y=5 So, now we know that at the point of intersection, the value of y = 5 The equation of the first line is 2x + 3y = 27 Subtituting for y with the value 5 in this equation, we get x = 6 So the answer is that the intersection is at the point (6, 5)
(4) (1, 2) ID : cn-9-linear-equations-in-two-variables [4] We are given the following: a. The equation is 5x + 3y = 11 b. The line has a point where the value of the ordinate is twice the value of the abscissa. The second fact implies that the point is of the form (x, 2x). Substituting y = 2x, in the equation 5x + 3y = 11 we get: 5x + 6x = 11 or, x = 1 Step 3 We have x = 1, which means the coordinates of the point will be (1, 2). (5) 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.
(6) y = d ID : cn-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
(7) y = -x - 1 ID : cn-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 -1. So c=-1 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 - (0))/(1 - (-1)) 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=-1 Now, if the line is a vertical one, then you can solve it by inspection. So the answer is y= -x - 1. (8) 2 We know the following facts - The equation of the line is 3x + b y = 22 - The point (4,5) lies on the line Substitute x=4 and y=5 in the equation 3 x 4 + b x 5 = 22 Solve this to find that the value of b is 2. (9) d. Natural Numbers A general equation in two variables has infinitely many solutions if there is no restriction placed on the values of the two variables (x and y here). However, it may have a unique solution if certain constraints are placed on it. Here we can see by observation that if x and y are constrained to be natural numbers, then it has a solution for x=y=1, and this is the only possible solution for natural numbers.
(10) c. 0 ID : cn-9-linear-equations-in-two-variables [7] 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 Step 3 Therefore, the graph of the equation y = mx + c will pass through the origin if value of c is 0 (11) a. (d, d) Try and trace the line x = y it the graph shown here You can see that any point on the line defined by the equation x = y will always have the value of x the same as y. Therefore a point on the line will have the form of (d, d) (12) a. y = 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 ((d, 0)). Out of the four options the only one it actually can match is y = 0
(13) c. y = 0 ID : cn-9-linear-equations-in-two-variables [8] Take a look at a graph below: We know that the value of y at all points on x-axis is zero, which means its equation should be y = 0. (14) b. 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. (15) c. parallel to y-axis at a distance b units from the origin If the equation for the line is x = b, this implies that the value of x is always b irrespective of the value of y. What this means is that this line is parallel to y-axis at a distance b units from the origin.