SOLUTION: A triangle is formed by the intersection of the lines 2x + 3y = 14
4X - 5y= - 16 and the x- axis.Find the area of the triangle
Thank you so much .
Question 664285: A triangle is formed by the intersection of the lines 2x + 3y = 14
4X - 5y= - 16 and the x- axis.Find the area of the triangle
Thank you so much . Answer by ReadingBoosters(3246) (Show Source):
You can put this solution on YOUR website! 2x+3y=14
as a linear equation
3y =-2x+14
y = -2/3x+14/3
and
4x-5y =-16
as a linear equation
5y = 4x+16
y = 4/5x + 16/5
...
y = -2/3x+14/3
y = 4/5x + 16/5
...
Graphically, we see the base of the triangle is from -4 and 7. The height of the triangle is 4. Area = 1/2bh = 1/2(11*4) = 22
...
Numerically,
To find the base you find the point where each line crosses the x axis and calculate the difference between those points.
y=-2/3x +14/3 crosses the x axis when y=0
0=-2/3x +14/3
2/3x=14/3
2/3x(3/2)=14/3(3/2)
x=7
The right bottom corner is at (7,0)
...
y=4/5x + 16/5
0=4/5x + 16/5
-4/5x = 16/5
-4/5x(5/4) = 16/5(-5/4)
x=-4
The left bottom corner is at (-4,0)
..
The distance between -4 and 0 is |-4| which is the same as 4 and the distance between 0 and 7 is |7| which is the same as 7. The distance between the x intercept points is 7+4=11. This is the base.
Alternatively, the difference in points is (7) -(-4)=11
...
The place where the line intersect is found by setting the two linear equations equal, as they share the y intersect point.
4/5x + 16/5 = -2/3x + 14/3
Isolate x
4/5x + 2/3x = 14/3 - 16/5
Multiply the entire expression by 15
12x + 10x = 70 - 48
22x = 22
x=1
The lines intersect where x = 1
...
Plug the x intersect of 1 back into the equation to find the y value at that intersection
y = -2/3(1) + 14/3 = 12/3 = 4
Double check with the other equation
y = 4/5(1) + 16/5 = 20/5 = 4
Both lines have a y point of 4 when x is 1
Therefore, they intersect at (1,4)
The y value indicates the height of the created triagle as 4
...
Area = 1/2bh
b=11
h=4
a = 1/2(11*4) = 1/2(44)= 22
.........................
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