Directions: Answer each question by creating and solving a system of equations to represent the situation presented. Show all work, including the system of equations.

1. Carpenter Charges

- Carpenter Charlie charges [tex]$15[/tex] per hour plus a flat rate of [tex]$100[/tex].
- Carpenter Chris charges [tex]$20[/tex] per hour plus a flat rate of [tex]$50[/tex].

For how many hours of work will the total charged be equal?

Let [tex]x[/tex] = the number of hours worked and [tex]y[/tex] = the total charge by the carpenter.

2. Basketball Team Rating System

- The Lions basketball team uses a rating system to rank players where they value each free throw for 1 point and each three-pointer for 3 points.
- John earned 100 points the day he shot a total of 60 free throws and three-pointers combined.

Let [tex]x[/tex] = free throws and [tex]y[/tex] = three-pointers.

How many of each shot did he make?

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.