Solving linear systems using technology

Question

The senior classes at high school A and high school B planned separate trips to the state fair. The senior class at high school A rented and filled 2 vans and 8 busses with 360 students. High school B rented and filled 6 vans and 1 bus with 114 students. Each van and each bus carried the same number of students. Find the number of students in each van and bus.

Answer 1

2V + 8b = 360

6v + 1 b = 114 --> b = 114-6v

2v + 8(114-6v) = 360

2v + 912 - 48v = 360

-46v + 912 = 360

-46v = 360-912

-46v = -552

v =12

then b = 114-6v = 114-6(12)

= 114-72

= 42

42 students on the bus

12 in the van...

Answer 2

x= # of students in van

y= # of students in bus

School A: 2x+8y=360

School B: 6x+y=114

You can use the substitution method

Substitution:

Isolate y for School B and substitute into School A

6x+y=114

y=-6x+114

Substitute into y-value of School A

2x+8y=360

2x+8(-6x+114)=360

Solve for x

2x-48x+912=360

-46x= -552

x=12 students in each van

Plug x back into either one of the original equations to solve for y.

6x+y=114

6(12)+y=114

72+y=114

y= 42 students in each bus

Answer 3

Lets say that the number of students in the van are represented by X and the number of students in the bus are represented by Y.

You would have the following equations:

high school A: 2x+8y=360

high school B: 6x+y=114

Now we can use the method of substitution to solve this system of equations. So I will solve the second equation for y and plug it into the first equation.

The second equation you would have to subtract 6x from both sides in order to get y by itself. So then you are left with

y=114-6x

Now we plug this into the first equation (2x+8y=360)

So you would get this equation: 2x+8(114-6x)=360

Distribute the 8 and you get: 2x+912-48x=360

Now combine the like terms (add the terms with the x's) and you get: -46x+912=360

Subtract 912 from both sides and you get: -46x=-552

Divide both sides by 46 and you get x=12 (this is the number of students in each van).

Now we can plug this value into one of the initial equations: I will use the equation for high school B (6x+y=114)

6*12+y=114

72+y=114, now subtract 72 from both sides and you get y=42 (this is the number of students in each bus)

Answer 4

You need to set up 2 equations then use system of equations to solve.

x= vans

y= buses

school A: 2x+8y=360

school B: 6x+1y=114

to solve you can use either substitution, graphing or elimination

Answer 5

First, we should decide what to call our variables before we try to solve any systems of linear equations based on a word problem. I think we should let v be the number of students that can fit in a van and let b be the number of students that can fit in a bus, but this is an arbitrary decision. You should decide on variable names that make sense to you.

In this case we know that 2v + 8b = 360 and 6v + b = 114. In order to solve this set of equations, we need to isolate a variable. We can do this by trying to multiply the equations by constants and adding them together in such a way that one variable cancels out and we isolate the other one. From experience, it seems easier to try and isolate y and cancel x here. If we multiply the first equation by -3 we get -3*(2v+8b) = -3*360 which implies -6v-24b = -1080. If we add this new, equivalent equation, to the first one, we get (-6v-24b)+(6v+b) = -1080+114 which implies -23b = -966. Dividing both sides by -23, we can see b = 42. To get the value of b, we can plug it into either of our equations. I think plugging it into the second will be easier (since we don't have to multiply b by 8), so 6v+42 = 114. Subtracting 42 from both sides, we get 6v = 72. Dividing both sides by 6, we get that v = 12. To sum up, each van can hold 12 students and each bus can hold 42 students.