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Find the equation of a line that passes through the intersection of x - y = 0 and 3x - 2y = 2 and forms a triangle at 1st Quadrant whose area is 9.
Solution:
To illustrate the problem, it is better to sketch the graph of two lines as follows:
For x - y = 0,
x - y = 0
y = x
slope (∆y/∆x), m = 1
y-intercept = 0
To trace the graph, plot 0 at the y-axis. This is your first point of the line (0, 0). Next, use the slope to get the second point. From the first point, count 1 unit to the right and then 1 unit upward.
For 3x - 2y = 2,
3x - 2y = 2
2y = 3x - 2
y = 3/2 x - 1
slope (∆y/∆x), m = 3/2
y-intercept = -1
To trace the graph, plot -1 at the y-axis. This is your first point of the line (0, -1). Next, use the slope to get the second point. From the first point, count 2 units to the right and then 3 units upward.
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| Photo by Math Principles in Everyday Life |
To get their point of intersection, we have to use the two given equations and solve for x and y as follows
from
Substitute the second equation to the first equation, we have
Substitute x to either of the two equations,
Therefore, their point of intersection is P(2, 2).
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| Photo by Math Principles in Everyday Life |
From the given word problem says "...forms a triangle at 1st Quadrant whose area is 9.", then the line that passes through point P will intersect at x-axis and y-axis. The resulting figure is a right triangle whose sides are x and y. The right angle is the origin while point P is located at the hypotenuse as follows
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| Photo by Math Principles in Everyday Life |
By using the formula for the area of a triangle, the first equation will be
By using the formula for getting the slope of two points, the second equation will be
Substitute the value of y from the first equation to the second equation, we have
Multiply both sides of the equation by x
If you will choose the positive sign,
Substitute the value of x either to the first equation or second equation, we have
The slope of a line is calculated as follows
Therefore, using the Point-Slope Form, the equation of a line is
If you will choose the negative sign,
Substitute the value of x either to the first equation or second equation, we have
The slope of a line is calculated as follows
Therefore, using the Point-Slope Form, the equation of a line is
