Question

write the linear equation that gives the rule for thiws table
X values: 3, 4 ,5,6
Y values: -5,-10,15,20

Answer 1

To find the linear equation that describes the relationship between the given \( X \) values and \( Y \) values, we can use the general form of a linear equation:

\[ Y = mX + b \]

where \( m \) is the slope and \( b \) is the y-intercept.

First, let's find the slope \( m \). The slope can be calculated using the formula:

\[ m = \frac{Y_2 - Y_1}{X_2 - X_1} \]

Let's use two points from the table to calculate the slope. Taking the points (3, -5) and (4, -10):

\[ m = \frac{-10 - (-5)}{4 - 3} = \frac{-10 + 5}{1} = \frac{-5}{1} = -5 \]

Now that we have the slope \( m = -5 \), we can use one of the points to find \( b \). Let's use the point (3, -5):

Using the equation \( Y = mX + b \):
\[
-5 = -5(3) + b
\]

\[
-5 = -15 + b
\]

\[
b = -5 + 15 = 10
\]

Now we have the slope \( m \) and the y-intercept \( b \):

The linear equation is:
\[
Y = -5X + 10
\]

You can verify this equation by substituting the \( X \) values from the table:

- For \( X=3 \): \( Y = -5(3) + 10 = -15 + 10 = -5 \) (Correct)
- For \( X=4 \): \( Y = -5(4) + 10 = -20 + 10 = -10 \) (Correct)
- For \( X=5 \): \( Y = -5(5) + 10 = -25 + 10 = -15 \) (Correct)
- For \( X=6 \): \( Y = -5(6) + 10 = -30 + 10 = -20 \) (Correct)

The rule for the given table of values is:
\[
Y = -5X + 10
\]