We will investigate the number of calories in ice cream and pie. We will define the variables as such:
[tex]\begin{gathered} x\text{ = Number of calories in pie} \\ y\text{ = Number of calories in ice-cream} \end{gathered}[/tex]From the given data a relationhsip between the number of calories in a pie ( x ) and the number of calories in an ice-cream scoop are given by the following statement:
" The number of calories in a piece of pie is 20 less than three times the number of calories in a scoop of ice cream. "
We will go ahead and decrypt the above given statement and express it mathematically in terms of declared variables ( x and y ) as follows:
[tex]\times\text{ = 3y - 20 }\ldots\textcolor{#FF7968}{Eq1}[/tex]The next statement gives us the combined calories in both ice-cream scoop and a pie:
" The pie and ice cream together have 500 calories. "
We will go ahead and decrypt the above given statement and express it mathematically in terms of declared variables ( x and y ) as follows:
[tex]x\text{ + y = 500 }\ldots\textcolor{#FF7968}{Eq2}[/tex]We have two equations ( derived ) and two variables ( x and y ):
[tex]\begin{gathered} \times\text{ - 3y = -20 }\ldots Eq1\text{ } \\ x\text{ + y = 500 }\ldots Eq2 \end{gathered}[/tex]We will solve the two equations ( Eq1 and Eq2 ) simultaneously using elimation method as follows:
[tex]\begin{gathered} x\text{ - 3y = -20 --- > }\text{\textcolor{#FF7968}{-x}}\text{ + 3y = 20} \\ \textcolor{#FF7968}{x}\text{ + y = 500} \\ =================== \\ 4y\text{ = 520 } \\ y\text{ = }\frac{520}{4} \\ \textcolor{#FF7968}{y}\text{\textcolor{#FF7968}{ = 130 calories}} \end{gathered}[/tex]We first multiplied the ( Eq1 ) by ( -1 ) then added the modified ( Eq1 ) to ( Eq2 ) whilst eliminating ( x ) and solved for the number of calories in an ice-cream scoop ( y ) i.e 130 calories.
Then we will back substitute the value of ( y ) into ( Eq2 ) and solve for ( x ):
[tex]\begin{gathered} x\text{ + y = 500} \\ x\text{ + 130 = 500} \\ \textcolor{#FF7968}{x}\text{\textcolor{#FF7968}{ = 370 calories}} \end{gathered}[/tex]Therefore, the number of calories in each confectionary is as such:
[tex]\begin{gathered} \textcolor{#FF7968}{x=370}\text{\textcolor{#FF7968}{ calories in pie}} \\ \textcolor{#FF7968}{y=130}\text{\textcolor{#FF7968}{ calories in an ice-cream scoop}} \end{gathered}[/tex]