1.
The linear equation 4x – 10y = 14 has:
Correct Answer
C. Infinitely many solutions
Explanation
The given linear equation has infinitely many solutions because it represents a straight line with a slope of 4/10 and a y-intercept of -14/10. This means that for every value of x, there will be a corresponding value of y that satisfies the equation. In other words, there are an infinite number of points that lie on the line and satisfy the equation.
2.
The equation 2x â€“ 3y = 5 has a unique solution if x and y are:
Correct Answer
A. Natural numbers
Explanation
The equation 2x â€“ 3y = 5 has a unique solution if x and y are natural numbers. This means that x and y must be positive integers. If x and y were any other type of number, such as positive real numbers, real numbers, or rational numbers, there would be an infinite number of solutions to the equation. Therefore, the only way for the equation to have a unique solution is if x and y are natural numbers.
3.
If (2, 0) is a solution of the linear equation 2x +3y = k, then the value of k is:
Correct Answer
A. 4
Explanation
Since (2, 0) is a solution to the equation 2x + 3y = k, we can substitute the values of x and y into the equation. Plugging in x = 2 and y = 0, we get 2(2) + 3(0) = k, which simplifies to 4 = k. Therefore, the value of k is 4.
4.
The graph of the linear equation 2x +3y = 6 cuts the y-axis at the point:
Correct Answer
D. (0, 2)
Explanation
The linear equation 2x + 3y = 6 can be rewritten as 3y = -2x + 6. To find the point where the graph cuts the y-axis, we set x = 0 and solve for y. Plugging in x = 0 into the equation, we get 3y = 6, which gives y = 2. Therefore, the graph cuts the y-axis at the point (0, 2).
5.
The equation y = 5, in two variables, can be written as:
Correct Answer
D. 0 . x + 1 . y = 5
Explanation
The equation y = 5 represents a linear equation in two variables, x and y. In this equation, the coefficient of x is 0, which means that x does not affect the value of y. On the other hand, the coefficient of y is 1, indicating that y is equal to 5. Therefore, the equation 0 . x + 1 . y = 5 correctly represents the given equation y = 5.
6.
Any point on the line y = x is of the form:
Correct Answer
D. (a, a)
Explanation
Any point on the line y = x has the same x and y coordinates, which means that the x-coordinate and y-coordinate are equal. Therefore, the correct answer is (a, a).
7.
The graph of x = 5 is a line:
Correct Answer
B. Parallel to the y-axis at a distance 5 units from the origin
Explanation
The correct answer is "Parallel to the y-axis at a distance 5 units from the origin." This is because the equation x = 5 represents a vertical line that is parallel to the y-axis. The line intersects the y-axis at the point (5,0) and is located 5 units to the right of the origin.
8.
x = 9, y = 4 is a solution of the linear equation:
Correct Answer
C. X + 2y = 17
Explanation
The given equation is x + 2y = 17. To check if x = 9 and y = 4 is a solution, we substitute these values into the equation.
9 + 2(4) = 17
9 + 8 = 17
17 = 17
Since the equation is true when x = 9 and y = 4, the answer x + 2y = 17 is correct.
9.
Any point on the x-axis is of the form:
Correct Answer
B. (x, 0)
Explanation
The correct answer is (x, 0) because any point on the x-axis has a y-coordinate of 0. The x-coordinate can be any real number, but the y-coordinate will always be 0. Therefore, the correct form for any point on the x-axis is (x, 0).
10.
If a linear equation has solutions (–3, 3), (0, 0), and (3, –3), then it is of the form:
Correct Answer
B. X + y = 0
Explanation
To find the form of the linear equation given the solutions (-3, 3), (0, 0), and (3, -3), we can use the point-slope form of a linear equation, which is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope of the line.
From the given solutions:
(-3, 3): x1 = -3, y1 = 3
(0, 0): x1 = 0, y1 = 0
(3, -3): x1 = 3, y1 = -3
First, let's find the slope using the points (-3, 3) and (0, 0): m = (y2 - y1) / (x2 - x1) m = (0 - 3) / (0 - (-3)) m = -3 / 3 m = -1
Now that we have the slope, we can plug it into the point-slope form for each point to determine the correct equation:
For point (-3, 3): y - 3 = -1(x - (-3)) y - 3 = -1(x + 3) y - 3 = -x - 3 y = -x
For point (0, 0): y - 0 = -1(x - 0) y = -x
For point (3, -3): y - (-3) = -1(x - 3) y + 3 = -x + 3 y = -x
Thus, the equation in the form that fits all the given points is: y = -x
Therefore, the correct option is: x + y = 0
11.
The graph of the linear equation 5x + 3y = 10 is a line that meets the x-axis at the point:
Correct Answer
C. (2, 0)
Explanation
The graph of a linear equation is a straight line. To find the point where the line intersects the x-axis, we need to find the x-coordinate of that point. We can do this by setting y=0 in the equation and solving for x. In this case, when y=0, the equation becomes 5x + 3(0) = 10, which simplifies to 5x = 10. Dividing both sides by 5 gives x = 2. Therefore, the point where the line meets the x-axis is (2, 0).
12.
The positive solutions of the equation ax + by + c = 0 always lie in the:
Correct Answer
A. Ist quadrant
Explanation
The positive solutions of the equation ax + by + c = 0 always lie in the first quadrant because in the first quadrant, both x and y values are positive. Since the equation is linear, the values of x and y can only be positive in order to satisfy the equation. Therefore, the positive solutions will always be in the first quadrant.
13.
The point of the form (a, –a) always lies on the line:
Correct Answer
D. X + y = 0
Explanation
The point (a, -a) always lies on the line x + y = 0 because when we substitute the value of x as a and the value of y as -a in the equation x + y = 0, we get a + (-a) = 0 which simplifies to 0 = 0. Therefore, the equation x + y = 0 holds true for the point (a, -a).
14.
The graph of x = 9 is a straight line:
Correct Answer
B. Parallel to y-axis
Explanation
The graph of x = 9 is a straight line parallel to the y-axis because the equation x = 9 means that the value of x is always 9 regardless of the value of y. This means that all points on the graph will have an x-coordinate of 9 and can have any y-coordinate. Therefore, the graph will be a vertical line parallel to the y-axis.
15.
The equation of the line parallel to the x-axis and six units above the origin is:
Correct Answer
C. Y = 6
Explanation
The equation of a line parallel to the x-axis means that the line will have a constant y-value regardless of the x-value. Since the line is six units above the origin, the y-value will be 6. Therefore, the equation of the line parallel to the x-axis and six units above the origin is y = 6.