To determine the missing value in the table while assuming a linear relationship, we need to first find the equation of the line that fits the given data points. The equation of a line can be expressed in slope-intercept form as:
[tex]\[ y = mx + b \][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. Let's break the process down into steps:
1. List known values:
- Given [tex]\( x \)[/tex] values: [tex]\( 1, 2, 3, 4 , 5 \)[/tex]
- Given [tex]\( y \)[/tex] values: [tex]\( -7, -5, -3, -1 \)[/tex]
2. Determine needed calculations:
- We need the slope [tex]\( m \)[/tex].
- We need the y-intercept [tex]\( b \)[/tex].
- Finally, we will use these to find [tex]\( y \)[/tex] for [tex]\( x = 5 \)[/tex].
3. Calculate the slope [tex]\( m \)[/tex]:
The slope [tex]\( m \)[/tex] is calculated by the formula:
[tex]\[
m = \frac{y_{n} - y_{1}}{x_{n} - x_{1}}
\][/tex]
Using the given points (1, -7) and (4, -1):
[tex]\[
m = \frac{(-1) - (-7)}{4 - 1} = \frac{-1 + 7}{3} = \frac{6}{3} = 2
\][/tex]
4. Find the y-intercept [tex]\( b \)[/tex]:
Substitute one pair of known [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values into the line equation:
Using the point (1, -7):
[tex]\[
-7 = 2(1) + b
\][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[
-7 = 2 + b \implies b = -7 - 2 \implies b = -9
\][/tex]
5. Form the complete equation of the line:
Using [tex]\( m = 2 \)[/tex] and [tex]\( b = -9 \)[/tex], the line equation is:
[tex]\[
y = 2x - 9
\][/tex]
6. Determine the missing [tex]\( y \)[/tex] value for [tex]\( x = 5 \)[/tex]:
Substitute [tex]\( x = 5 \)[/tex] into the line equation:
[tex]\[
y = 2(5) - 9 = 10 - 9 = 1
\][/tex]
Therefore, the missing value for [tex]\( y \)[/tex] when [tex]\( x = 5 \)[/tex] is [tex]\( \boxed{1} \)[/tex].
