Question

Luca and William are playing a video game and they have scored a total of 20,00
points. Luca scored 1,00
more points than William. If you let l=
the number of points that Luca scored, and w=
the number of points that William scored, then the problem can be represented by the system:

l+w=20,00o and l=w+1,00

Graph the system. How many points did each boy score?

Answer 1

To solve the problem, let's define the system of equations based on the information given:

1. \( l + w = 2000 \) (Equation 1)
2. \( l = w + 100 \) (Equation 2)

### Step 1: Substitute Equation 2 into Equation 1

Substituting Equation 2 into Equation 1, we replace \( l \) with \( w + 100 \):

\[
(w + 100) + w = 2000
\]

### Step 2: Simplify

Combine the terms:

\[
2w + 100 = 2000
\]

### Step 3: Solve for \( w \)

Subtract 100 from both sides:

\[
2w = 1900
\]

Now divide by 2:

\[
w = 950
\]

### Step 4: Use \( w \) to find \( l \)

Now substitute \( w = 950 \) back into Equation 2:

\[
l = w + 100 = 950 + 100 = 1050
\]

### Conclusion

Luca scored \( 1050 \) points and William scored \( 950 \) points.

### Summary of Points

- Luca: \( 1050 \)
- William: \( 950 \)

### Graphing the System

To graph the system, we'll rearrange both equations:

**Equation 1:** \( l + w = 2000 \) can be rewritten as \( w = 2000 - l \).

**Equation 2:** \( l = w + 100 \) can be rewritten as \( w = l - 100 \).

Now, we can plot these lines on a graph:

1. For \( w = 2000 - l \):
- If \( l = 0 \), then \( w = 2000 \).
- If \( l = 2000 \), then \( w = 0 \).

2. For \( w = l - 100 \):
- If \( l = 100 \), then \( w = 0 \).
- If \( l = 200 \), then \( w = 100 \).

The intersection of these lines will give the solution \( (l, w) = (1050, 950) \).

### Graphically representing the points

- The first line starts at (0, 2000) and ends at (2000, 0).
- The second line starts at (100, 0) and has a slope of 1.

The point of intersection at \( (1050, 950) \) confirms the scores of Luca and William.