To determine the equation of the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] given the data in the table, we need to recognize the pattern and derive a linear function of the form [tex]\( y = mx + c \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( c \)[/tex] is the y-intercept.
We begin by examining the changes in the [tex]\( y \)[/tex]-values for each corresponding [tex]\( x \)[/tex]-value:
- For [tex]\( x = 0 \)[/tex], [tex]\( y = 3 \)[/tex]
- For [tex]\( x = 1 \)[/tex], [tex]\( y = 5 \)[/tex]
- For [tex]\( x = 2 \)[/tex], [tex]\( y = 7 \)[/tex]
- For [tex]\( x = 3 \)[/tex], [tex]\( y = 9 \)[/tex]
- For [tex]\( x = 4 \)[/tex], [tex]\( y = 11 \)[/tex]
- For [tex]\( x = 5 \)[/tex], [tex]\( y = 13 \)[/tex]
1. Finding the slope [tex]\( m \)[/tex]:
The slope [tex]\( m \)[/tex] is defined as the change in [tex]\( y \)[/tex] divided by the change in [tex]\( x \)[/tex] between any two points. For example, between [tex]\( (0, 3) \)[/tex] and [tex]\( (1, 5) \)[/tex]:
[tex]\[
m = \frac{y_1 - y_0}{x_1 - x_0} = \frac{5 - 3}{1 - 0} = \frac{2}{1} = 2
\][/tex]
Checking this with another pair, for consistency, using points [tex]\( (1, 5) \)[/tex] and [tex]\( (2, 7) \)[/tex]:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - 5}{2 - 1} = \frac{2}{1} = 2
\][/tex]
The slope is consistently 2.
2. Finding the y-intercept [tex]\( c \)[/tex]:
To find the y-intercept [tex]\( c \)[/tex], we substitute one of the known points and the slope into the equation [tex]\( y = mx + c \)[/tex]. Using the point [tex]\( (0, 3) \)[/tex]:
[tex]\[
3 = 2 \cdot 0 + c \implies c = 3
\][/tex]
3. The complete equation:
The relationship can thus be expressed as:
[tex]\[
y = 2x + 3
\][/tex]
4. Finding [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex]:
Substituting [tex]\( x = 0 \)[/tex] into the equation:
[tex]\[
y = 2 \cdot 0 + 3 = 3
\][/tex]
Therefore, the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex] is [tex]\( \boxed{3} \)[/tex].
