Yesterday, Esperanza worked for 6 hours and earned $120. Today, Esperanza worked 8 hours and earned $156. Write the equation of the linear function that models the relationship between the number of hours Esperanza works and the amount of money she earns. Enter your answer in y = mx + b form.
To find the equation of the linear function, let's first define the variables:
Let x be the number of hours Esperanza works.
Let y be the amount of money Esperanza earns.
We are given two data points:
For yesterday: x = 6 and y = 120.
For today: x = 8 and y = 156.
To find the slope (m) of the linear function, we use the formula:
m = (change in y) / (change in x)
m = (156 - 120) / (8 - 6)
m = 36 / 2
m = 18
Now, let's substitute one of the data points and the slope into the slope-intercept form of the equation (y = mx + b) to find the value of b.
Using the data point (x = 6, y = 120):
120 = 18(6) + b
120 = 108 + b
b = 120 - 108
b = 12
Therefore, the equation of the linear function that models the relationship between the number of hours Esperanza works and the amount of money she earns is:
y = 18x + 12
We can use the slope-intercept form of a linear function, y = mx + b, where y represents the amount of money Esperanza earns, x represents the number of hours she works, m represents the rate of pay per hour, and b represents the y-intercept (the amount of money she would earn if she didn't work at all).
To find the rate of pay per hour, we can use the information from the two data points we have:
- Yesterday, Esperanza worked for 6 hours and earned $120, so her rate of pay per hour is $20 (120/6).
- Today, Esperanza worked for 8 hours and earned $156, so her rate of pay per hour is $19.50 (156/8).
We can use either of these rates of pay per hour to write the equation of the linear function, but we'll use the average rate of pay per hour, which is $19.75 ((20 + 19.5) / 2).
So our equation is:
y = 19.75x + b
To find the y-intercept, we can use either of the data points we have. Let's use yesterday's:
120 = 19.75(6) + b
Simplifying and solving for b, we get:
b = 0.25
So our final equation is:
y = 19.75x + 0.25
This equation models the relationship between the number of hours Esperanza works and the amount of money she earns.