Question

The equations 20x +12y = 140 and 45x +20y = 280 represent the cost for lunch and dinner for a family eating out on vacation. If x is the number of adults and y is the number of children, how many adults are in the family?425

Answer 1

Given the equations:

`\begin{gathered} 20x+12y=140\text{ \lparen1\rparen} \\ 45x+20y=280\text{ \lparen2\rparen} \end{gathered}`

Where,

x= Number of adults.

y= Number of children.

Isolating y in the equation (1):

`\begin{gathered} 20x+12y=140 \\ 12y=140-20x \\ y=(140-20x)/(12)\text{ \lparen3\rparen} \end{gathered}`

Replacing the equation (3) in equation (2).

`\begin{gathered} 45x+20y=280 \\ 45x+20*((140-20x)/(12))=280 \end{gathered}`

Simplifying:

`\begin{gathered} 45x+(20*140-20*20x)/(12)=280 \\ \\ 45x+(2800-400x)/(12)=280 \\ \\ (45*12x+2800-400x)/(12)=280 \end{gathered}`

`540x+2800-400x=12*280`

Finally, solving for x:

`\begin{gathered} 140x=3360-2800 \\ 140x=560 \\ x=(560)/(140)=(56)/(14)=4 \end{gathered}`

The number the adults is x=4.

And the number of children is:

`\begin{gathered} y=(140-20(4))/(12)=5 \\ \end{gathered}`