Question

A group of 202 people went on the overnight camping trip taking 60 tents with them some of the tents held 2 people each and the rest held 4 people each all the tents were filled to capacity and every person got to sleep in a tent how many of each type of tent were taken on the camping trip

Answer 1

ThePart A

Let "x" be the number of tents that can hold 4 people and "y" represent the number of tents that can hold 4 people.

They bought a total of 60 tents, you can write an expression for the total number of tents as follows:

`x+y=60`

The group is of 202 people if each tent was filled to capacity and everyone had a place to sleep, you can express the number of people that slept in tents as follows:

`4x+2y=202`

These equations represent the system of equations.

Part B

Next, find the intercepts of both lines to be able to graph them:

-First equation:

y-intercept:

Replace the equation with x=0

`\begin{gathered} x+y=60 \\ 0+y=60 \\ y=60 \end{gathered}`

The coordinates are (0,60)

x-intercetpt:

Replace the equation with y=0

`\begin{gathered} x+y=60 \\ x+0=60 \\ x=60 \end{gathered}`

The coordinates are (60,0)

- Second equation:

y-intercept:

Replace the equation with x=0 and solve for y:

`\begin{gathered} 4x+2y=202 \\ 4\cdot0+2y=202 \\ 2y=202 \\ (2y)/(2)=(202)/(2) \\ y=101 \end{gathered}`

The coordinates are (0,101)

x-intercept:

Replace the equation with y=0 and solve for x:

`\begin{gathered} 4x+2y=202 \\ 4x+2\cdot0=202 \\ 4x=202 \\ (4x)/(4)=(202)/(4) \\ x=50.5 \end{gathered}`

The coordinates are (50.0,)

Plot the 5 points and draw the lines:

Part C

The solution of the system is the point where both lines intersect, first, solve the equation system:

`\begin{gathered} x+y=60 \\ 4x+2y=202 \end{gathered}`

-Solve the first equation for y:

`\begin{gathered} x+y=60 \\ y=60-x \end{gathered}`

-Substitute the expression in the second equation

`4x+2(60-x)=202`

Distribute the multiplication on the parentheses term:

`\begin{gathered} 4x+2\cdot60+2\cdot(-x)=202 \\ 4x+120-2x=202 \\ 4x-2x+120=202 \\ 2x+120=202 \end{gathered}`

Subtract 120 to both sides of the equal sign

`\begin{gathered} 2x+120-120=202-120 \\ 2x=82 \end{gathered}`

Divide both sides by 2

`\begin{gathered} (2x)/(2)=(82)/(2) \\ x=41 \end{gathered}`

- Replace the value of x in the expression obtained for y:

`\begin{gathered} y=60-x \\ y=60-41 \\ y=19 \end{gathered}`

The solution of the equation system is (41,19)

Part D

The solution of the equation system is x= 41 and y= 19, this means that they took 41 tents that could hold 4 people and 19 tents that could hold 2 people.