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:
[tex]x+y=60[/tex]
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:
[tex]4x+2y=202[/tex]
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
[tex]\begin{gathered} x+y=60 \\ 0+y=60 \\ y=60 \end{gathered}[/tex]
The coordinates are (0,60)
x-intercetpt:
Replace the equation with y=0
[tex]\begin{gathered} x+y=60 \\ x+0=60 \\ x=60 \end{gathered}[/tex]
The coordinates are (60,0)
- Second equation:
y-intercept:
Replace the equation with x=0 and solve for y:
[tex]\begin{gathered} 4x+2y=202 \\ 4\cdot0+2y=202 \\ 2y=202 \\ \frac{2y}{2}=\frac{202}{2} \\ y=101 \end{gathered}[/tex]
The coordinates are (0,101)
x-intercept:
Replace the equation with y=0 and solve for x:
[tex]\begin{gathered} 4x+2y=202 \\ 4x+2\cdot0=202 \\ 4x=202 \\ \frac{4x}{4}=\frac{202}{4} \\ x=50.5 \end{gathered}[/tex]
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:
[tex]\begin{gathered} x+y=60 \\ 4x+2y=202 \end{gathered}[/tex]
-Solve the first equation for y:
[tex]\begin{gathered} x+y=60 \\ y=60-x \end{gathered}[/tex]
-Substitute the expression in the second equation
[tex]4x+2(60-x)=202[/tex]
Distribute the multiplication on the parentheses term:
[tex]\begin{gathered} 4x+2\cdot60+2\cdot(-x)=202 \\ 4x+120-2x=202 \\ 4x-2x+120=202 \\ 2x+120=202 \end{gathered}[/tex]
Subtract 120 to both sides of the equal sign
[tex]\begin{gathered} 2x+120-120=202-120 \\ 2x=82 \end{gathered}[/tex]
Divide both sides by 2
[tex]\begin{gathered} \frac{2x}{2}=\frac{82}{2} \\ x=41 \end{gathered}[/tex]
- Replace the value of x in the expression obtained for y:
[tex]\begin{gathered} y=60-x \\ y=60-41 \\ y=19 \end{gathered}[/tex]
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.
