Q32. - Algebra 2

Question

At the 2000 Summer Olympics in Sydney, Australia, the United States won 97 medals. 

They won 6 more gold medals than bronze and 8 fewer silver medals than bronze.

How can you determine the number and type of medals U.S. Olympians won?

Include the following in your answer:

A demonstration of how to find the number of each type of medal won by the U.S. Olympians, and

A description of another situation where you can use a system of three equations in three variables to solve a problem..

Step-by-Step Solution

Answer

The U.S. Olympians won, 39 gold medals, 33 bronze medals, and 25 silver medals.

Step 1 – First construct a system of equations corresponding to the given situation.

Let $g$be the number of gold medals, $s$ be the number of silver medals, and $b$be the number of bronze medals.

 In the figure, it is given that the total 97 medals were won.

So, $g+s+b=97$.

 In the figure, it is given that they won 6 more gold medals than the bronze.

So,

$g=b+6\Rightarrow g-b=6$

 In the figure, it is given that they won 8 fewer silver medals than the bronze.

This can be mathematically written as $s=b-8\Rightarrow s-b=-8$.

 So, the system of equations will be:

$\begin{array}{c}g+s+b=97\\ g-b=6\\ s-b=-8\end{array}$

Step 2 – Use the elimination method to get the system of equations in two variables.

Add $g+s+b=97$ to the equation $g-b=6$.

$\begin{array}{l}\underset{\_}{\begin{array}{l}g+s+b=97\\ g-b=6\end{array}}\\ 2g+s+0=103\end{array}$

So, the resultant equation is $2g+s=103$.

 Add $g+s+b=97$ to the equation $s-b=-8$

$\begin{array}{l}\underset{\_}{\begin{array}{l}g+s+b=97\\ s-b=-8\end{array}}\\ 9+2s+0=89\end{array}$

So, the resultant equation is $g+2s=89$.

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Step 3 – Solve the system of equations in two variables

Multiply$g+2s=89$ by 2 and subtract the new resultant equation from $2g+s=103$

$\begin{array}{l}\underset{\_}{\begin{array}{c}g+2s=89\\ 2g+s=103\end{array}}\begin{array}{c}\text{multiply by 2}\\ \end{array}\underset{\_}{\begin{array}{l}2g+4s=178\\ (-)2g+s=103\end{array}}\\ 0+3s=75\end{array}$

Solve$3s=75$ for $s$:

$\begin{array}{c}3s=75\\ \frac{3s}{3}=\frac{75}{3}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} divide both sides by }3\\ s=25\end{array}$

Step 4 – Find the values of g and b .

Substitute $s=25 in s-b=-8$ and find the value of $b$

$\begin{array}{c}s-b=-8\\ 25-b=-8\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Substitute 2}5\text{ for }s\\ -b=-33\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Subtract }25\text{ from both sides}\\ b=33\end{array}$

Substitute $b=33 in g-b=6$ and find the value of $g$

$\begin{array}{c}g-b=6\\ g-33=6\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Substitute }33\text{ for }b\\ g=39\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Subtract 33 from both sides}\end{array}$

.Hence, the solution of the given system of equations is$\left(g,b,s\right)=\left(39,33,25\right)$.

So, the U.S. Olympians won, 39 gold medals, 33 bronze medals, and 25 silver medals.

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