What is the area of the triangle formed by lines y = 5-x, 2y

Question

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

A. 7.5
B. 8.0
C. 9.0
D. 10.5
E. 15.0

M21-33

The vertices of the triangle formed are (0,0), (5,0) and (2,3)

The first two vertices are fine, but how do we arrive at the third vertice of (2,3)? Since the equation for the third line is y = 0x + 0 / y = 0 it should be a horizontal line. Look forward to any explanations.

Bunuel · Accepted Answer

We have three equations of lines:  y_1=0  (x-axis),  y_2=\frac{3x}{2} ,  y_3= 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:  y_1=0=y_2=\frac{3x}{2}  -->  x=0 ,  y=0  -->  vertex_1=(0,0) ;
Second vertex:  y_1=0=y_3=5-x  -->  x=5 ,  y=0  -->  vertex_2=(5,0) ;
Third vertex:  y_2=\frac{3x}{2}=y_3=5-x  -->  x=2 ,  y=3  -->  vertex_3=(2,3) .

Base=5 ,  Height=3  -->  Area=\frac{1}{2}*Base*Height=7.5 .

Answer: A.

abhi758 · Answer

Thanks Bunnel for the detailed explanation! Very helpful..

vigneshpandi · Answer

I got the three co-ordinates as (5,5) (0,0) and (2,3)...
Anyone Please tel me where am i going wrong?

Bunuel · Answer

Second vertex: intersection of y=0 and y=5-x --> y_1=0=y_3=5-x --> x=5 , y=0 --> vertex_2=(5,0) ; Or refer to the graphs below: graph.php.png Hope it's clear.

medanova · Answer

Bunuel, what do you use for making the graphs?

Bunuel · Answer

There are many online graph plotters, try this one: https://graph-plotter.cours-de-math.eu/ or: https://www.wolframalpha.com/

sanjoo · Answer

Bunuel.. I got this question in gmat club tests..

What is the area of a triangle enclosed by line 2x+3y=6, line y−x=2 and X-axis on the coordinate plane?

My question is.. why approach is different ?

If 2x+3y=6 and x=0 then y=2. And when y=0 then x=3.. (0,2) (3,0)

then y cant we do same with y=5-x? (5,0) (0,5)

and 2x=3y.. x=0 and y=0

where m wrong?

Bunuel · Answer

Below is that question. Please point out what differences are you talking about. What is the area of a triangle enclosed by line 2x+3y=6, line y-x=2 and X-axis on the coordinate plane? A. 3 B. 4 C. 5 D. 8 E. 10 Look at the diagram below: Area.png Lines y=-\frac{2}{3}x+2 and y=x+2 intersect at point (0, 2). So the height of enclosed triangle is 2. Next, X-intercept of line y=-\frac{2}{3}x+2 is (3, 0) and X-intercept of line y=x+2 is (-2, 0), so the base of enclosed triangle is 3-(-2)=5. The area is \frac{1}{2}*base*height=\frac{1}{2}*5*2=5 . Answer: C.

sanjoo · Answer

actually this highlighted thing is my question.. y have we taken another equation y=3/2x to get the points of equation y=5-x?

What is the area of a triangle enclosed by line 2x+3y=6, line y-x=2 and X-axis on the coordinate plane?

and i tuk this question (in bold text) to show u that , we didnt do same thing in this question. we tuk different vertices by individully solving an equation.

i tried my best to explain my question

thank u

Bunuel · Answer

The approaches are identical for both questions:

The base in both, is on x-axis. Find x-intercepts of other two lines and find the difference between them to get the length of the base.
Next, find at what point do these two lines intersect. y-coordinate of that point would be the height.

To get intersection point of two lines y=3x/2 and y=5-x, we equate 3x/2 and 5-x to get x-coordinate of intersection point --> 3x/2=5-x --> x=2. Then we substitute it into either of the equations to get y=coordinate of intersection point --> y=5-2=3. So, y=3x/2 and y=5-x intersect at (2,3).

Hope it's clear.

rxn · Answer

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

Having trouble understanding the best approach to calculating vertices of the triangle which are given as follows:

(0,0)
(5,0)
(2,3)

Can anyone explain the best approach for calculating these? I'm fine with calculating the area once I have the vertices.

Thanks.

ind23 · Answer

The vertices are the points where these lines intersect with each other. so, solve two two equation together: point where y=0 and 2y = 3x intersect: put y=0 in 2y = 2x => x = 0, so coordinates (x,y) = (0,0) point where y = 0 and y = 5 - x intersect: put y=0 in y = 5-x => x = 5, so coordinates (x,y) = (5,0) point where 2y = 3x and y = 5 - x intersect: put y=3x/2 in y = 5-x => x = 2; y = 3, so coordinates (x,y) = (2,3) -------------------------------- Kudos if the answer helped

martinnotceoyet · Answer

Alternatively, you could draw a quick graph on your scratch paper using the equations. For 2y= 3x , use the equation y= 3x/2 to draw the line.
Then you just have to read the coordinates to have the coordinates of the vertices.
I still think ind23's method is faster, but it is important to visualize geometry problems as it makes it easier to answer.

Bunuel · Answer

Merging similar topics. Please refer to the discussion above. P.S. Please read carefully and follow: rules-for-posting-please-read-this-before-posting-133935.html Pay attention to rules 1, 2, 3, 7 and 8. Thank you.

OreoShake · Answer

Hi Bunuel,

Please help me solidify the understanding for this - the reason you equated the line equations is because intersecting lines have the same slope right? Thank you.

Bunuel · Answer

No, we are equating to get the x-intercept of the point of intersection.

dave13 · Answer

hi Bunuel i really dont get your solution so we have this y = 5-x, 2y = 3x, and y = 0 and we have these coordinaty geometry map with x and y axis y = 5-x, so i plugged in this 2y = 3x --> x = 2y/3 into this y = 5-x and got y = 3 so i have y= 3 and x =2 hence y is height and x is basis ---> y = 3 and x = 2 i am sure you can visualize my wring solution

area (3*2)/2 = 3

pushpitkc - help pls:) why are solving in a totally different way:? are quant gmatclub tests harder than the quant on gmat exam ?

NandishSS · Answer

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

a. 7.5
b. 8.0
c. 9.0
d 10.5
e. 15.0

For line 2 
2y = 3x ==> y=3/2x

If x = 0, y=3/2(0) ==> 0
y=0 , x=0

Where Am'I going wrong?  chetan2u ,  VeritasPrepKarishma  ,  Bunuel  ,  mikemcgarry

KarishmaB · Answer

I am not sure why you think you are going wrong.

Given 2y = 3x, we get y = (3/2)x + 0 (to put it in the y = mx + c form)
This just means that it is a line with slope 1.5 and y intercept 0. So it must pass through the origin (which you got as - if x = 0 then y = 0)

y = 0 is the equation of the x axis. So we know that these two lines intersect at the origin. So it is one of the three vertices of the triangle.

Similarly, now we find where the third line intersects these two lines to get the other two vertices of the triangle.

What is the area of the triangle formed by lines y = 5-x, 2y

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What is the area of the triangle formed by lines y = 5-x, 2y

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

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From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards