What is the Area of the Triangle Formed by Lines y = 5-x, 2y GMAT Problem Solving

Question: What is the area of the triangle formed by lines y = 5-x, 2y = 3x, and y = 0?

1. 7.5
2. 8.0
3. 9.0
4. 10.5
5. 15.0

Correct Answer: A
 

Solution and Explanation 
Approach Solution 1:
Given to us, there are three lines that form a triangle. Those are:
Y = 5 - x, 2y = 3x, and y = 0.
It is asked to find out the area of the triangle.
Firstly let’s check the slopes of the sides of the triangle.
For y = 5 - x
M = -1
For 2y = 3x
M = 3/2
For y = 0
M = 0
If no two product of slopes is -1,therefore
This is not a right-angled triangle.
Equating every two equations to get the vertex of triangle.
Putting y = 0 in y = 5-x
0 = 5-x
X = 5
One point is (5,0)
Putting y = 0 in 2y = 3x
3x = 0
X = 0
Second point is (0,0)
Putting y = 5-x in 2y=3x
2(5-x) = 3x
10 - 2x = 3x
5x = 10
X = 2
Putting x = 2 in 1st equation we get,
Y = 5-x
Y = 5-2
Y = 3
Third point is (2,3)
Now it can be observed that the value of y for the points (5,0) and (0,0) is 0.
Therefore both of these points lie on the x-axis.
The base will be x2 - x1
Base = 5- 0 = 5
The third point is (2,3) which does not lie on x axis rather it lies above the x-axis
To find the height of the triangle we have to check the value of y.
For (2,3) the height is 3 as it lies 3 units above the x-axis.
So we have base = 5 units
Height = 3 units
Area of a triangle = \(\frac{1}{2}\)* base * height
Area = \(\frac{1}{2}\) * 5 * 3
Area = 15/2 = 7.5sq units

Approach Solution 2:
We have three equations of lines: y1= 0(x-axis), y2= 3x/2, y3=5−x.

Equate the functions to get the x-coordinate of the vertex (x-coordinate of the intersection point) and then substitute this value in either of functions to get y-coordinate of the vertex.

First vertex: y1=0=y2=3x/2--> x=0, y=0y=0 --> vertex1=(0,0)vertex1=(0,0);
Second vertex: y1=0=y3=5−x --> x=5, y=0 --> vertex2=(5,0)vertex2=(5,0);
Third vertex: y2=3x/2=y3=5−x-->x=2,y=3→ vertex3=(2,3)

Base=5, Height=3 --> Area=1/2∗Base∗Height=7.5Area=12∗Base∗Height=7.5.

“What is the area of the triangle formed by lines y = 5-x, 2y”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Official Guide 2018 Quantitative Review".To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. GMAT Quant practice papers improve the mathematical knowledge of the candidates as it represents multiple sorts of quantitative problems.

 

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