To determine the equation that represents the relationship between the amount borrowed [tex]\( x \)[/tex] and the amount repaid [tex]\( y \)[/tex], we follow these steps:
1. Identify the data points:
We have the following pairs of values from the given table:
[tex]\[
(100, 135), (200, 260), (500, 635), (800, 1010)
\][/tex]
2. Assume a linear relationship:
We assume that the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] can be described by a linear equation of the form:
[tex]\[
y = mx + b
\][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
3. Calculate the slope [tex]\( m \)[/tex]:
The slope [tex]\( m \)[/tex] represents the rate of change of [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex]. Based on our calculations:
[tex]\[
m = 1.25
\][/tex]
4. Calculate the y-intercept [tex]\( b \)[/tex]:
The y-intercept [tex]\( b \)[/tex] represents the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex]. Based on our calculations:
[tex]\[
b = 10
\][/tex]
5. Write the equation:
Substituting the values of [tex]\( m \)[/tex] and [tex]\( b \)[/tex] into the linear equation [tex]\( y = mx + b \)[/tex], we get:
[tex]\[
y = 1.25x + 10
\][/tex]
Therefore, the equation that represents the relationship between the amount borrowed [tex]\( x \)[/tex] and the amount repaid [tex]\( y \)[/tex] is:
[tex]\[
y = 1.25x + 10
\][/tex]
