To find the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex], we are looking for a linear equation of the form:
[tex]\[ y = mx + b \][/tex]
1. First, we identify two points from the given data: (0, -1) and (1, 3).
2. We calculate the slope [tex]\( m \)[/tex] using these two points. The slope [tex]\( m \)[/tex] is given by:
[tex]\[ m = \frac{(y_2 - y_1)}{(x_2 - x_1)} \][/tex]
For the points (0, -1) and (1, 3):
[tex]\[ x_1 = 0, \, y_1 = -1 \][/tex]
[tex]\[ x_2 = 1, \, y_2 = 3 \][/tex]
So,
[tex]\[ m = \frac{3 - (-1)}{1 - 0} = \frac{3 + 1}{1 - 0} = \frac{4}{1} = 4 \][/tex]
3. Next, we calculate the y-intercept [tex]\( b \)[/tex]. We can use the slope-intercept form of the line equation [tex]\( y = mx + b \)[/tex] and one of the given points to find [tex]\( b \)[/tex]. Using the point (0, -1):
[tex]\[ y_1 = mx_1 + b \][/tex]
[tex]\[ -1 = 4 \cdot 0 + b \][/tex]
Therefore,
[tex]\[ b = -1 \][/tex]
4. Combining the slope and y-intercept we found:
[tex]\[ y = 4x - 1 \][/tex]
Thus, the linear relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is:
[tex]\[ y = 4x - 1 \][/tex]
