Question

[x] [ya] [yb]
[0] [0] [3]
[1] [2] [3.5]
[2] [4] [4]

what linear equation is represented by the table of values?

Answer 1

Two students were testing the parachute designs they made for an egg-drop project. Both students dropped their eggs from a height of 10 feet. One design made the egg fall at a rate 5 feet per second, and the other design made the egg fall at a rate of 3 feet per second. If x represents the time in seconds that the egg falls and y represents the height of the egg, which system of equations represents the situation described?

Answer 2

To find the linear equation represented by the given table of values, we can first identify the variables from the table. Let's denote \( x \) as the first column (which seems to be your independent variable) and \( y \) as the second column (your dependent variable).

From the table:
- When \( x = 0 \), \( y = 0 \)
- When \( x = 1 \), \( y = 2 \)
- When \( x = 2 \), \( y = 4 \)

Now, we can determine the linear equation in the form of \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

1. **Finding the slope (m):**
We can calculate the slope \( m \) using two points from the table, for instance, \( (0, 0) \) and \( (1, 2) \).
\[
m = \frac{\Delta y}{\Delta x} = \frac{2 - 0}{1 - 0} = 2
\]

2. **Finding the y-intercept (b):**
From the table, we see that when \( x = 0 \), \( y = 0 \), which means the y-intercept \( b = 0 \).

Now we can write the equation based on the slope and y-intercept:
\[
y = 2x + 0 \quad \text{or simply} \quad y = 2x
\]

However, notice that there are two values \( [yb] \) in the table which influence the interpretation. Given this setup, it seems additional context might define the second value. To confirm an overall view, if you'd like to analyze the relationship between multiple outputs (like \( ya \) and \( yb \)), you might consider:

1. \( ya \) values
- When \( x = 0 \), \( ya = 0 \)
- When \( x = 1 \), \( ya = 2 \)
- When \( x = 2 \), \( ya = 4 \)

This directly corresponds with our previously evaluated \( y = 2x \).

2. **Regarding \( yb \):**
The table shows \( y = 3, 3.5, 4 \) against each x. The relationship seems to vary. To infer, you may create a second equation analyzing \( yb \).

However, if the focus rests on one line, we conclude based on \( ya \) that the linear equation is:
\[
\boxed{y = 2x}
\]

Answer 3

To represent the situation with a system of equations, we need to formulate the equations based on the height of the egg at a given time \( x \).

### Scenario Breakdown

1. **Height of the Egg:** The starting height from which the eggs are dropped is 10 feet.

2. **Fall Rate:**
- For the first design, the egg falls at a rate of 5 feet per second.
- For the second design, the egg falls at a rate of 3 feet per second.

### Equations for Each Design

1. **First Design (5 feet per second):**
The height \( y_1 \) of the egg for the first design as a function of time \( x \) can be expressed as:
\[
y_1 = 10 - 5x
\]
This equation reflects the fact that for every second that passes, the height decreases by 5 feet.

2. **Second Design (3 feet per second):**
The height \( y_2 \) of the egg for the second design as a function of time \( x \) can be expressed as:
\[
y_2 = 10 - 3x
\]
Similarly, this equation indicates that the height decreases by 3 feet for each second that passes.

### System of Equations

Combining these two equations, we obtain the system of equations that represents the situation:

\[
\begin{cases}
y_1 = 10 - 5x \quad (1) \\
y_2 = 10 - 3x \quad (2)
\end{cases}
\]

### Conclusion

Thus, the system of equations representing the height of the eggs for both designs over time \( x \) is:

\[
\begin{cases}
y = 10 - 5x \quad \text{(first design)} \\
y = 10 - 3x \quad \text{(second design)}
\end{cases}
\]

This clearly outlines how each design affects the height of the egg over time as it falls.