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.
### 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.
