Sure! Let's break down the process to find the missing \( y \) value when \( x = 5 \).
### Step-by-Step Solution:
1. List the Given Values:
We have the following pairs \((x, y)\):
- \( x = 1, y = 16 \)
- \( x = 2, y = 25 \)
- \( x = 3, y = 34 \)
- \( x = 4, y = 43 \)
- \( x = 5, y = ? \)
2. Calculate the Differences in \( x \) and \( y \):
- The differences in \( x \), also known as \( \Delta x \), are consistent due to equal spacing: \( \Delta x = 2 - 1 = 3 - 2 = 4 - 3 = 5 - 4 = 1 \).
- The differences in \( y \), also known as \( \Delta y \), can be calculated:
[tex]\[
\Delta y_1 = 25 - 16 = 9
\][/tex]
[tex]\[
\Delta y_2 = 34 - 25 = 9
\][/tex]
[tex]\[
\Delta y_3 = 43 - 34 = 9
\][/tex]
3. Determine the Average Rate of Change (Slope \( m \)):
- Since all \( \Delta y \) values are the same, the slope \( m \) between sequential \( y \) values is the same:
[tex]\[
m = \frac{\Delta y}{\Delta x} = \frac{9}{1} = 9
\][/tex]
4. Find the Equation of the Line:
- We know the slope \( m = 9 \) and a point on the line \( (x, y) = (4, 43) \).
- Using the point-slope form of the linear equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept:
[tex]\[
y = mx + b
\][/tex]
- Plugging in the values \( y = 43 \), \( x = 4 \), and \( m = 9 \):
[tex]\[
43 = 9 \cdot 4 + b
\][/tex]
[tex]\[
43 = 36 + b
\][/tex]
[tex]\[
b = 43 - 36 = 7
\][/tex]
5. Using the Equation to Find the Missing \( y \):
- Now we have the equation of the line \( y = 9x + 7 \).
- To find \( y \) when \( x = 5 \):
[tex]\[
y = 9 \cdot 5 + 7
\][/tex]
[tex]\[
y = 45 + 7
\][/tex]
[tex]\[
y = 52
\][/tex]
Thus, the missing [tex]\( y \)[/tex] value for [tex]\( x = 5 \)[/tex] is [tex]\( \boxed{52} \)[/tex]. The slopes between sequential [tex]\( y \)[/tex] values were consistently [tex]\( 9.0 \)[/tex].
