In a coordinate plane, the points (2,4) and (3,-1) are on a line.
Which of the following must be true?
1. The line crosses the x-axis.
2. The line passes through (0, 0).
3. The line stays above the x-axis at all times.
4. The line rises from the lower left to the upper right.
5. The line is parallel to the y-axis.
Since the y of the first point is above the x-axis and the y of the second point is below, the line must surely cross the x-axis
To determine which statements must be true, let's analyze the given points (2,4) and (3,-1) on a coordinate plane.
1. The line crosses the x-axis:
To determine if the line crosses the x-axis, we need to check if any of the given points have a y-coordinate of 0. Since neither (2,4) nor (3,-1) have a y-coordinate of 0, the line does not necessarily cross the x-axis. Therefore, statement 1 does not have to be true.
2. The line passes through (0, 0):
To determine if the line passes through (0, 0), we need to check if the equation of the line passing through (2,4) and (3,-1) also passes through (0, 0). To do this, we can calculate the slope of the line.
Slope (m) = (change in y) / (change in x)
= (-1 - 4) / (3 - 2)
= -5
Using the slope-intercept form of a linear equation (y = mx + b), we can substitute one of the given points to solve for the y-intercept (b). Let's use point (2,4):
4 = -5(2) + b
4 = -10 + b
b = 14
Therefore, the equation of the line passing through (2,4) and (3,-1) is y = -5x + 14. Substituting (0,0) into the equation, we get:
0 = -5(0) + 14
0 = 0 + 14
0 = 14
Since 0 does not equal 14, the line does not pass through (0,0). Therefore, statement 2 does not have to be true.
3. The line stays above the x-axis at all times:
To determine if the line stays above the x-axis at all times, we need to check if both given points have a y-coordinate greater than 0. Since both (2,4) and (3,-1) have y-coordinates greater than 0, the line stays above the x-axis at all times. Therefore, statement 3 must be true.
4. The line rises from the lower left to the upper right:
To determine if the line rises from the lower left to the upper right, we can compare the y-coordinates of the two given points. Since the y-coordinate of the second point (-1) is less than the y-coordinate of the first point (4), the line does not rise from the lower left to the upper right. Therefore, statement 4 does not have to be true.
5. The line is parallel to the y-axis:
To determine if the line is parallel to the y-axis, we can compare the x-coordinates of the two given points. Since the x-coordinate of the second point (3) is different from the x-coordinate of the first point (2), the line is not parallel to the y-axis. Therefore, statement 5 does not have to be true.
In conclusion, the statements that must be true are:
3. The line stays above the x-axis at all times.
To determine which of the statements must be true, we can analyze the properties of the given points (2, 4) and (3, -1) on a coordinate plane.
1. The line crosses the x-axis:
To check if the line crosses the x-axis, we need to see if there is a point on the line where the y-coordinate is zero. Since the point (2, 4) has a positive y-coordinate and the point (3, -1) has a negative y-coordinate, it means the line does not pass through y = 0 on the x-axis. Therefore, statement 1 is not necessarily true.
2. The line passes through (0, 0):
To determine if the line passes through the origin (0, 0), we need to see if it satisfies the equation y = mx + b, where m is the slope of the line and b is the y-intercept. However, we do not have enough information about the line's equation or slope to determine if it passes through the origin. Therefore, statement 2 is not necessarily true.
3. The line stays above the x-axis at all times:
To determine if the line stays above the x-axis, we need to check if all the y-coordinates of the points on the line are positive. From the given points, it can be seen that the point (3, -1) has a negative y-coordinate, which means the line dips below the x-axis at some point. Therefore, statement 3 is not necessarily true.
4. The line rises from the lower left to the upper right:
To determine if the line rises from the lower left to the upper right, we need to check if the slope of the line is positive. The slope of a line can be calculated using the formula:
slope (m) = (change in y) / (change in x)
For the given points (2, 4) and (3, -1):
slope (m) = (4 - (-1)) / (2 - 3)
= 5 / (-1)
= -5
Since the slope is negative (-5), it means the line falls from the upper left to the lower right, not rising from the lower left to the upper right. Therefore, statement 4 is not true.
5. The line is parallel to the y-axis:
To determine if the line is parallel to the y-axis, we need to check if all the x-coordinates of the points on the line are the same. Looking at the given points (2, 4) and (3, -1), we can see that the x-coordinates are different, which means the line is not parallel to the y-axis. Therefore, statement 5 is not true.
In conclusion, none of the provided statements are necessarily true based on the given points (2, 4) and (3, -1).