Question

write the linear equation that gives rule for this table
x y
3 -5
-10
4 -15
5 -20
6

Answer 1

To find the linear equation that represents the relationship between \( x \) and \( y \) in the given table, we first identify the values we can see:

- When \( x = 3 \), \( y = -5 \)
- When \( x = 4 \), \( y = -15 \)
- When \( x = 5 \), \( y = -20 \)
- When \( x = 6 \), the value is missing.

### Step 1: Find the Slope

First, let’s calculate the slope \( m \) based on the provided points. We can use the points \( (3, -5) \) and \( (4, -15) \):

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-15 - (-5)}{4 - 3} = \frac{-15 + 5}{4 - 3} = \frac{-10}{1} = -10
\]

### Step 2: Use the Point-Slope Form

Now that we have the slope \( m = -10 \), we can use the point-slope form of a linear equation, which is:

\[
y - y_1 = m(x - x_1)
\]

Using the point \( (3, -5) \):

\[
y - (-5) = -10(x - 3)
\]
\[
y + 5 = -10(x - 3)
\]
\[
y + 5 = -10x + 30
\]
\[
y = -10x + 30 - 5
\]
\[
y = -10x + 25
\]

### Conclusion

The linear equation that models the relationship between \( x \) and \( y \) is:

\[
y = -10x + 25
\]

### Verification with Other Points:

Let’s verify this with another point from the table, \( (4, -15) \):

\[
y = -10(4) + 25 = -40 + 25 = -15
\]

And check with \( (5, -20) \):

\[
y = -10(5) + 25 = -50 + 25 = -25
\]

It appears there's a slight discrepancy here. The value I expected to find is -20, which suggests that the slope might need to be reevaluated based on more data points.

Calculating the slope between different pairs of points:

- Between \( (4, -15) \) and \( (5, -20) \):
\[
m = \frac{-20 - (-15)}{5 - 4} = \frac{-20 + 15}{1} = -5
\]

So there seems to be a consistent slope of -5. Adjusting for that:

Using the point \( (4, -15) \):
\[
y + 15 = -5(x - 4)
\]
\[
y + 15 = -5x + 20
\]
\[
y = -5x + 5
\]

### Correct Linear Equation:
The corrected linear equation based on a consistent slope of -5 from the points is:

\[
y = -5x + 5
\]

This model matches the observed data more accurately given multiple points in the table.