Answer :
Sure, let's go through the solution step-by-step:
a. Linear Equation in Standard Form for Three Unknowns
We are told that the United States won a total of 104 gold, silver, and bronze medals in the 2012 Summer Olympics. This can be represented by the linear equation:
[tex]\[ g + s + b = 104 \][/tex]
where [tex]\( g \)[/tex] is the number of gold medals, [tex]\( s \)[/tex] is the number of silver medals, and [tex]\( b \)[/tex] is the number of bronze medals.
b. Relationship Between the Number of Silver and Bronze Medals
We are given in part (b) of the problem that the United States won 46 gold medals, and the same number of silver and bronze medals. This implies that the number of silver medals [tex]\( s \)[/tex] is equal to the number of bronze medals [tex]\( b \)[/tex]. Therefore, the relationship can be expressed as:
[tex]\[ s = b \][/tex]
c. Solving the Linear Equation
Given that there are 46 gold medals and [tex]\( s = b \)[/tex], we use this information to solve for the number of silver and bronze medals:
1. Substitute the gold medals into the equation from part (a): [tex]\( g = 46 \)[/tex].
2. The equation becomes [tex]\( 46 + s + b = 104 \)[/tex].
3. Since [tex]\( s = b \)[/tex], we can replace [tex]\( b \)[/tex] with [tex]\( s \)[/tex] in the equation:
[tex]\[ 46 + s + s = 104 \][/tex]
4. Simplify the equation:
[tex]\[ 46 + 2s = 104 \][/tex]
5. Solve for [tex]\( s \)[/tex]:
[tex]\[ 2s = 104 - 46 \][/tex]
[tex]\[ 2s = 58 \][/tex]
[tex]\[ s = 29 \][/tex]
6. Since [tex]\( s = b \)[/tex], we also have [tex]\( b = 29 \)[/tex].
So, the final medal count is:
- Gold medals, [tex]\( g = 46 \)[/tex]
- Silver medals, [tex]\( s = 29 \)[/tex]
- Bronze medals, [tex]\( b = 29 \)[/tex]
This gives us the complete solution for the number of each type of medal won by the United States.
a. Linear Equation in Standard Form for Three Unknowns
We are told that the United States won a total of 104 gold, silver, and bronze medals in the 2012 Summer Olympics. This can be represented by the linear equation:
[tex]\[ g + s + b = 104 \][/tex]
where [tex]\( g \)[/tex] is the number of gold medals, [tex]\( s \)[/tex] is the number of silver medals, and [tex]\( b \)[/tex] is the number of bronze medals.
b. Relationship Between the Number of Silver and Bronze Medals
We are given in part (b) of the problem that the United States won 46 gold medals, and the same number of silver and bronze medals. This implies that the number of silver medals [tex]\( s \)[/tex] is equal to the number of bronze medals [tex]\( b \)[/tex]. Therefore, the relationship can be expressed as:
[tex]\[ s = b \][/tex]
c. Solving the Linear Equation
Given that there are 46 gold medals and [tex]\( s = b \)[/tex], we use this information to solve for the number of silver and bronze medals:
1. Substitute the gold medals into the equation from part (a): [tex]\( g = 46 \)[/tex].
2. The equation becomes [tex]\( 46 + s + b = 104 \)[/tex].
3. Since [tex]\( s = b \)[/tex], we can replace [tex]\( b \)[/tex] with [tex]\( s \)[/tex] in the equation:
[tex]\[ 46 + s + s = 104 \][/tex]
4. Simplify the equation:
[tex]\[ 46 + 2s = 104 \][/tex]
5. Solve for [tex]\( s \)[/tex]:
[tex]\[ 2s = 104 - 46 \][/tex]
[tex]\[ 2s = 58 \][/tex]
[tex]\[ s = 29 \][/tex]
6. Since [tex]\( s = b \)[/tex], we also have [tex]\( b = 29 \)[/tex].
So, the final medal count is:
- Gold medals, [tex]\( g = 46 \)[/tex]
- Silver medals, [tex]\( s = 29 \)[/tex]
- Bronze medals, [tex]\( b = 29 \)[/tex]
This gives us the complete solution for the number of each type of medal won by the United States.