College

The United States won 104 gold (g), silver (s), and bronze (b) medals in the 2012 Summer Olympics.

a. Select the linear equation in standard form for three unknowns.

b. The United States won 46 gold medals and the same number each of silver and bronze medals. Select the relationship between the number of silver to bronze medals in an equation of two unknowns.

c. With the information given in b, solve the linear equation in a for the number of gold, silver, and bronze medals won.

Options:

A. [tex]g + s + b = 104[/tex]

B. [tex]g = 29, \quad s = 46, \quad b = 29[/tex]

C. [tex]g = b[/tex]

D. [tex]s = b[/tex]

E. [tex]g = 46, \quad s = 29, \quad b = 29[/tex]

F. [tex]g + s + b = 100[/tex]

Answer :

Sure! Let's go through the solution step by step:

### a. Formulating the Linear Equation

We start by expressing the total number of medals won by the United States as a linear equation. We have:
- [tex]\( g \)[/tex] representing the number of gold medals,
- [tex]\( s \)[/tex] representing the number of silver medals, and
- [tex]\( b \)[/tex] representing the number of bronze medals.

The total number of medals won is 104. Therefore, our equation in standard form becomes:

[tex]\[ g + s + b = 104 \][/tex]

### b. Using Additional Information

We know that the United States won 46 gold medals and the same number of silver and bronze medals. This information can be translated into an equation as follows:

[tex]\[ s = b \][/tex]

This means the number of silver medals is equal to the number of bronze medals.

### c. Solving the Equation

From the given information in part b, we substitute [tex]\( g = 46 \)[/tex] into the original equation:

[tex]\[ 46 + s + b = 104 \][/tex]

Using the relationship [tex]\( s = b \)[/tex], we can substitute [tex]\( b \)[/tex] in the equation:

[tex]\[ 46 + b + b = 104 \][/tex]

Simplifying this:

[tex]\[ 46 + 2b = 104 \][/tex]

We can solve for [tex]\( b \)[/tex] by subtracting 46 from both sides:

[tex]\[ 2b = 104 - 46 \][/tex]

[tex]\[ 2b = 58 \][/tex]

Next, we divide by 2 to find [tex]\( b \)[/tex]:

[tex]\[ b = 29 \][/tex]

Since [tex]\( s = b \)[/tex], it follows:

[tex]\[ s = 29 \][/tex]

### Results

With these calculations, we find:
- Gold medals ([tex]\( g \)[/tex]) = 46
- Silver medals ([tex]\( s \)[/tex]) = 29
- Bronze medals ([tex]\( b \)[/tex]) = 29

These results match the conditions given in the problem, confirming our solution.