Answer :
Sure! Let's solve these questions step by step.
### Problem 1: Carpenter Charges
We have two carpenters: Charlie and Chris. Let's set up the equations based on their charges:
- Charlie charges: [tex]$15 per hour plus a flat rate of $[/tex]100.
- Equation: [tex]\( y = 15x + 100 \)[/tex]
- Chris charges: [tex]$20 per hour plus a flat rate of $[/tex]50.
- Equation: [tex]\( y = 20x + 50 \)[/tex]
We want to find the number of hours ([tex]\( x \)[/tex]) at which their total charges ([tex]\( y \)[/tex]) are equal. So, we need to set their charge equations equal to each other:
[tex]\[ 15x + 100 = 20x + 50 \][/tex]
Now, let's solve for [tex]\( x \)[/tex]:
1. Subtract [tex]\( 15x \)[/tex] from both sides:
[tex]\[ 100 = 5x + 50 \][/tex]
2. Subtract 50 from both sides:
[tex]\[ 50 = 5x \][/tex]
3. Divide both sides by 5:
[tex]\[ x = 10 \][/tex]
So, for 10 hours of work, both Charlie and Chris will charge the same amount.
### Problem 2: Basketball Team Rating
John's performance is measured using the following information:
- Free throws are worth: 1 point each.
- Three-pointers are worth: 3 points each.
- John scored a total of 100 points with a total of 60 shots (free throws + three-pointers).
Let:
- [tex]\( x \)[/tex] be the number of free throws.
- [tex]\( y \)[/tex] be the number of three-pointers.
We can form the following equations:
1. Points equation: [tex]\( 1x + 3y = 100 \)[/tex]
2. Shot equation: [tex]\( x + y = 60 \)[/tex]
Let's solve this system of equations:
1. From the second equation, express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[ x = 60 - y \][/tex]
2. Substitute [tex]\( x \)[/tex] in the first equation:
[tex]\[ 1(60 - y) + 3y = 100 \][/tex]
3. Simplify and solve for [tex]\( y \)[/tex]:
[tex]\[ 60 - y + 3y = 100 \][/tex]
[tex]\[ 60 + 2y = 100 \][/tex]
[tex]\[ 2y = 100 - 60 \][/tex]
[tex]\[ 2y = 40 \][/tex]
[tex]\[ y = 20 \][/tex]
4. Substitute [tex]\( y = 20 \)[/tex] back into the equation for [tex]\( x \)[/tex]:
[tex]\[ x + 20 = 60 \][/tex]
[tex]\[ x = 60 - 20 \][/tex]
[tex]\[ x = 40 \][/tex]
Therefore, John made 40 free throws and 20 three-pointers.
### Problem 1: Carpenter Charges
We have two carpenters: Charlie and Chris. Let's set up the equations based on their charges:
- Charlie charges: [tex]$15 per hour plus a flat rate of $[/tex]100.
- Equation: [tex]\( y = 15x + 100 \)[/tex]
- Chris charges: [tex]$20 per hour plus a flat rate of $[/tex]50.
- Equation: [tex]\( y = 20x + 50 \)[/tex]
We want to find the number of hours ([tex]\( x \)[/tex]) at which their total charges ([tex]\( y \)[/tex]) are equal. So, we need to set their charge equations equal to each other:
[tex]\[ 15x + 100 = 20x + 50 \][/tex]
Now, let's solve for [tex]\( x \)[/tex]:
1. Subtract [tex]\( 15x \)[/tex] from both sides:
[tex]\[ 100 = 5x + 50 \][/tex]
2. Subtract 50 from both sides:
[tex]\[ 50 = 5x \][/tex]
3. Divide both sides by 5:
[tex]\[ x = 10 \][/tex]
So, for 10 hours of work, both Charlie and Chris will charge the same amount.
### Problem 2: Basketball Team Rating
John's performance is measured using the following information:
- Free throws are worth: 1 point each.
- Three-pointers are worth: 3 points each.
- John scored a total of 100 points with a total of 60 shots (free throws + three-pointers).
Let:
- [tex]\( x \)[/tex] be the number of free throws.
- [tex]\( y \)[/tex] be the number of three-pointers.
We can form the following equations:
1. Points equation: [tex]\( 1x + 3y = 100 \)[/tex]
2. Shot equation: [tex]\( x + y = 60 \)[/tex]
Let's solve this system of equations:
1. From the second equation, express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[ x = 60 - y \][/tex]
2. Substitute [tex]\( x \)[/tex] in the first equation:
[tex]\[ 1(60 - y) + 3y = 100 \][/tex]
3. Simplify and solve for [tex]\( y \)[/tex]:
[tex]\[ 60 - y + 3y = 100 \][/tex]
[tex]\[ 60 + 2y = 100 \][/tex]
[tex]\[ 2y = 100 - 60 \][/tex]
[tex]\[ 2y = 40 \][/tex]
[tex]\[ y = 20 \][/tex]
4. Substitute [tex]\( y = 20 \)[/tex] back into the equation for [tex]\( x \)[/tex]:
[tex]\[ x + 20 = 60 \][/tex]
[tex]\[ x = 60 - 20 \][/tex]
[tex]\[ x = 40 \][/tex]
Therefore, John made 40 free throws and 20 three-pointers.