A video game requires at least 4 points to advance. Each solved puzzle is worth two points. Each solved riddle is worth 1 point. If x is the number of solved puzzles and y is the number of solved riddles, which graph represents the overall equation represented by this scenario (all points may not apply to the scenario)?

Question

On a coordinate plane, a solid straight line has a negative slope and goes through (0, 2) and (4, 0). Everything below the line is shaded.
On a coordinate plane, a solid straight line has a negative slope and goes through (0, 2) and (4, 0). Everything above the line is shaded.
On a coordinate plane, a solid straight line has a negative slope and goes through (0, 4) and (2, 0). Everything to the left of the line is shaded.
On a coordinate plane, a solid straight line has a negative slope and goes through (0, 4) and (2, 0). Everything to the right of the line is shaded.

Answer 1

To solve this problem, let's first formulate the required conditions based on the points system described.

1. Each solved puzzle (x) gives 2 points.
2. Each solved riddle (y) gives 1 point.
3. To advance, the total points scored must be at least 4 points.

The equation based on this scenario can be formed as follows:

\[ 2x + y \geq 4 \]

This means that the line corresponding to the equation \( 2x + y = 4 \) defines the boundary, and everything above this line will satisfy the condition \( 2x + y \geq 4 \).

Now, let's rewrite the equation in a form suitable for graphing:

1. Solve for y: \[ y = 4 - 2x \]

The y-intercept here is 4 (when \( x = 0 \)), and the x-intercept is 2 (when \( y = 0 \)).

Now that we have the intercepts, we can summarize the line's equation and its properties:

• The line \( y = 4 - 2x \) passes through points (0, 4) and (2, 0).
• Since we need to satisfy the inequality \( 2x + y \geq 4 \), we should shade the area above this line.

Now, let's go through the options provided:

1. First Option: A line going through (0, 2) and (4, 0), shaded below.
2. Second Option: A line going through (0, 2) and (4, 0), shaded above.
3. Third Option: A line going through (0, 4) and (2, 0), shaded to the left.
4. Fourth Option: A line going through (0, 4) and (2, 0), shaded to the right.

The correct representation of \( 2x + y \geq 4 \) is the third option, which shows a line going through (0, 4) and (2, 0) with shading to the right.

Therefore, the answer is:

The third option: On a coordinate plane, a solid straight line has a negative slope and goes through (0, 4) and (2, 0). Everything to the right of the line is shaded.