Need help starting systems of linear equations.

[Iamthatis]

I am in Algebra 2 and I need help setting up these problems:

Question #1
--"Three intersecting landing runways at an airport are described by 9y + 45 =7x, 12x = 35 - 5y, and y - 3x = 2. Plotting these runways on a map, an architect determines the number of square units of area enclosed by the intersections of these runways. What is the amount of enclosed area?"

My problem with #1
-I looked at the above problem and graphed it, using slope-intercept. However, I realized that my teacher is trying to teach me to set up equations, and the graph will not show exact measurements anyways. I need help with this problem in getting it set up, and the beginning of solving it. My teacher taught us how to do some basic problems, but to be honest, I have never been taught how to do something like this.

Question #2
--"Flim has $X, Flam has $Y, and Flum has $Z. Altogether they have $a. Flim has c times as many dollars as Flam, and the sum of Flim's and Flam's dollars is b times as much as Flum's dollars. How much money does each of them have."

My problem with Question #2
-Basically, the same applies for this question as #1. The teacher did not explain how to deal with a problem like this. I could solve it if there was a true numerical value somewhere in here, but there isn't. I would greatly appreciate seeing how this is set up, and the first few steps in solving it.

If someone would help me understand these questions, it would be VERY greatly appreciated.

 

[pka]

This is a nice set of problems. If one does them, that person will learn a lot.
What have you done on any of these?
Many of us already know how to do them. We can help you learn to do them.
But it is not our policy (with one notable exception) to simply do them for you.
Please show us what you have done towards working them.

 

[Iamthatis]

This is all I have gotten for the second one
a=X+Y+Z
x=(Y*C)
(X+Y)*b=Z
After plugging that in I got:
a=(Y*C)+Y+[(X+Y)*b]
... and that's pretty much it for that problem

On the first one, like I said, I cannot even start that problem due to the fact that I don't understand how I would go about finding the area given lines on a graph.

 

[Deleted member 4993]

I am in Algebra 2 and I need help setting up these problems:

Question #1
--"Three intersecting landing runways at an airport are described by 9y + 45 =7x, 12x = 35 - 5y, and y - 3x = 2. Plotting these runways on a map, an architect determines the number of square units of area enclosed by the intersections of these runways. What is the amount of enclosed area?"

My problem with #1
-I looked at the above problem and graphed it, using slope-intercept. However, I realized that my teacher is trying to teach me to set up equations, and the graph will not show exact measurements anyways. I need help with this problem in getting it set up, and the beginning of solving it. My teacher taught us how to do some basic problems, but to be honest, I have never been taught how to do something like this.

sketching is actually very good idea.

What you need to do is this:

Find the points of intersections - taking pair of lines.

Those are the vertices of your triangle.

Now you can find the length of each side.

You can use Heron's formula to find the area.


Question #2
--"Flim has $X, Flam has $Y, and Flum has $Z. Altogether they have $a. Flim has c times as many dollars as Flam, and the sum of Flim's and Flam's dollars is b times as much as Flum's dollars. How much money does each of them have."

My problem with Question #2
-Basically, the same applies for this question as #1. The teacher did not explain how to deal with a problem like this. I could solve it if there was a true numerical value somewhere in here, but there isn't. I would greatly appreciate seeing how this is set up, and the first few steps in solving it.

If someone would help me understand these questions, it would be VERY greatly appreciated.

so:

X + Y + Z = a...Altogether they have $a.....(1)

X = c* Y Flim has c times as many dollars as Flam .......(2)

X + Y = b * Z...the sum of Flim's and Flam's dollars is b times as much as Flum's dollars......(3)

Using (2) in (3)

c*Y + Y = b*Z

Z = (1+c)/b * Y.................................(4)

Using 2 & 4 in 1

c*Y + Y + (1+c)/b * Y = a...................(5)

Now solve for 'Y' from (5) and continue...

 

[Iamthatis]

I appreciate your attempt to help, but I am still very much so on the confused side. I do not know how to find the point of intersection (besides graphing, which is not a whole number) and the "C*Y+Y+(1+C)/b*Y=a" still leaves me clueless. I still do not how to solve that for Y.

 

[stapel]

I do not know how to find the point of intersection....

The intersection of any two lines in the solution to the system of equations formed by the equations for the two lines. So, to find the intersection point, solve the system of equations.

It is often simplest, in this sort of context, to solve each line equation for "y=", and then set the other halves equal to each other. For instance, the intersection of 3x + 2y = 6 and x - 4y = 8 would by found by:

. . . . .3x + 2y = 6
. . . . .2y = -3x + 6
. . . . .y = -(3/2)x + 3

. . . . .x - 4y = 8
. . . . .x - 8 = 4y
. . . . .(1/4)x - 2 = y

. . . . .-(3/2)x + 3 = (1/4)x - 2
. . . . .-6x + 12 = x - 8
. . . . .12 + 8 = x + 6x
. . . . .20 = 7x
. . . . .20/7 = x

Then back-solve for y:

. . . . .y = -(3/2)[20/7] + 3
. . . . .y = -(30/7) + 21/7
. . . . .y = -9/7

So the intersection of the two lines would be (x, y) = (20/7, -9/7).

Follow the same process with your line equations.

Eliz.

 

[Loren]

Prob #1.
Solve the equations two at a time simultaneously. For instance...

9y+45=7x
y=3x+2

-7x+9y=-45
9x- 3y= -6

-7x+9y=-45
27x-9y=-8

20x = -63
x=(-63/20) or -3.15
y=3(-63/20)+2=-149/20 or -7.45

one of the vertices of the triangle is at (-63/20,-149/20)

Now, in like manner find the other two vertices.

Once you have the three points, use the distance between two points formula to find the length of each side of the triangle. That formula is...

\(\displaystyle d=\sqrt{(y_2-y_1)^2+(x_2-x_1)^2}\)

After you have the length of each side, use Hero's (Heron's) formula to find the area...

\(\displaystyle A=\sqrt{s(s-a)(s-b)(s-c)}\) where s=(a+b+c)/2.

Good luck. This must be an extra credit problem???

 

[Iamthatis]

Thank you, but sadly no, this is not an extra credit problem. It is part of a graded homework. Thank you, I am pretty confident with #1. However, I am still unable to solve
"c*Y + Y + (1+c)/b * Y = a" that
I have never been taught to deal with such a complex equation.

 

[Deleted member 4993]

Thank you, but sadly no, this is not an extra credit problem. It is part of a graded homework. Thank you, I am pretty confident with #1. However, I am still unable to solve

"c*Y + Y + (1+c)/b * Y = a"

Factor out "Y" on the Left-hand-side.

Just like in the following equation:

5.2x + x + 4.8 x = 22

x * (5.2 + 1 + 4.8) = 22

11 * x = 22

x = 22/11 = 2

Follow the same procedure....


that
I have never been taught to deal with such a complex equation.

 

[Iamthatis]

I can add and multiply NUMBERS without a problem. It is trying to eliminate some of these variables that kills me.


*Edit*
Okay, I think I managed to solve #2
My answers are:
X=C*Y
Z=(1+C)/B*Y
Y=b*Z+X
Is this correct?
~This is hopeless... I'm going to fail. The sad thing is, that this homework counts 60 points on our next test. I have solved two problems for 15 points. I understand HOW to set up the problems, it is just that once they are set up, the method for solving this has not yet been taught. For example, Heron's formula is required in one, and I haven't the faintest idea what it is.

 

[stapel]

I can add and multiply NUMBERS without a problem. It is trying to eliminate some of these variables that kills me.... the method for solving this has not yet been taught.

To learn how to solve systems of linear equations, try here.

Okay, I think I managed to solve #2
My answers are:
X=C*Y
Z=(1+C)/B*Y
Y=b*Z+X
Is this correct?

The idea, I expect, is to solve the system in terms of "x=", "y=", and "z=".

Directly from the problem statement, you have the following system:

. . . . .(1) x + y + z = a

. . . . .(2) x = cy

. . . . .(3) x + y = bz

Plug (2) in for "x" in (3) to get:

. . . . .cy + y = bz

. . . . .(c + 1)y = bz

. . . . .(4) [(c + 1)y] / b = z

Now plug (2) and (4) into (1) to get:

. . . . .cy + y + [(c + 1)y] / b = a

. . . . .(bcy)/b + (by)/b + [(c + 1)y]/b = a

. . . . .[bcy + by + (c + 1)y]/b = a

. . . . .bcy + by + (c + 1)y = ab

. . . . .(bc + b + c + 1)y = ab

Solve for "y=". Then work backwards to find x and z in terms only of a and b.

Heron's formula is required in one, and I haven't the faintest idea what it is.

Heron's Formula is one method for solving the exercise. It is not the only one. Another method would be to draw the three lines, find their intersection points, and then start dividing the area into triangles having the slanty lines and various horizontal and vertical lines for sides.

You know the formula for the area A of a triangle, given the height h and the base b. With that information and the judicious drawing of triangles, you can find the total enclosed area. :wink:

Eliz.

 

[Iamthatis]

... Still confused, this is due tomorrow... so I'll just have to try to get as many right as possible and hope for the best I guess. If I ace the written part, that is at least a 40 then. I guess one B on a report card won't kill me, but I won't appreciate it. I hate asking for the answers, but if you understand that the teacher REQUIRES me to show my work, you understand that in the long run, the answers won't do me any good.

 

[Deleted member 4993]

... Still confused, this is due tomorrow... so I'll just have to try to get as many right as possible and hope for the best I guess. If I ace the written part, that is at least a 40 then. I guess one B on a report card won't kill me, but I won't appreciate it. I hate asking for the answers, but if you understand that

the teacher REQUIRES me to show my work,

This is a SUPERB teacher - I would like to congratulate the teacher

Like pka said - these are excellent problems - require some work - we have shown you most of the steps.

You started the post with header "Need help starting....". We went far beyond that - just short of giving you the final answer.

Now you need to EARN your grade by finishing these.




you understand that in the long run, the answers won't do me any good.