Problem 1 :
The table shows the temperature of a fish tank during an experiment. Write the appropriate linear equation to find the temperature at any time.
Solution :
Step 1 :
Notice that the change in the temperature is the same for each increase of 1 hour in time. So, the relationship is linear.
Step 2 :
Let x stand for time and y stand for temperature.
Choose any two points in the form (x, y), from the table to find the slope :
For example, let us choose (0, 82) and (1, 80).
Use the slope formula.
m = (y_{2} - y_{1})/(x_{2} - x_{1})
Substitute (0, 82) for (x_{1}, y_{1}) and (1, 80) for (x_{2}, y_{2}).
m = (80 - 82)/(1 - 0)
m = -2/1
m = -2
Step 3 :
Find the y-intercept using the slope and any point from the table.
Slope-intercept form equation of a line :
y = mx + b
Substitute m = -2, and (x, y) = (0, 82).
82 = -2(0) + b
82 = 0 + b
82 = b
Step 4 :
Now, substitute m = -2 and b = 82 in slope-intercept form equation of a line.
y = mx + b
y = -2x + 82
Problem 2 :
Elizabeth’s cell phone plan lets her choose how many minutes are included each month. The table shows the plan’s monthly cost y for a given number of included minutes x. Write an equation in slope-intercept form to represent the situation.
Solution :
Step 1 :
Notice that the change in cost is the same for each increase of 100 minutes. So, the relationship is linear.
Step 2 :
Choose any two points in the form (x, y), from the table to find the slope :
For example, let us choose (100, 14) and (200, 20).
Use the slope formula.
m = (y_{2} - y_{1})/(x_{2} - x_{1})
Substitute (100, 14) for (x_{1}, y_{1}) and (200, 20) for (x_{2}, y_{2}).
m = (20 - 14)/(200 - 100)
m = 6/100
m = 0.06
Step 3 :
Find the y-intercept using the slope and any point from the table.
Slope-intercept form equation of a line :
y = mx + b
Substitute m = 0.06, and (x, y) = (100, 14).
14 = 0.06(100) + b
14 = 6 + b
8 = b
Step 4 :
Now, substitute m = 0.06 and b = 8 in slope-intercept form equation of a line.
y = mx + b
y = 0.06x + 8
Problem 3 :
The rent charged for space in an office building is a linear relationship related to the size of the space rented. At west main street office rentals, $750 rent charged for 600 square feet of space and $1150 rent charged for 900 square feet of space. Write an equation in slope-intercept form for the rent at West Main Street Office Rentals.
Solution :
Step 1 :
Identify the independent and dependent variables.
The independent variable (x) is the square footage of floor space.
The dependent variable (y) is the monthly rent.
Step 2 :
Write the information given in the problem as ordered pairs.
The rent for 600 square feet of floor space is $750 :
(600, 750)
The rent for 900 square feet of floor space is $1150 :
(900, 1150)
Step 3 :
Find the slope.
m = (y_{2} - y_{1})/(x_{2} - x_{1})
Substitute (600, 750) for (x_{1}, y_{1}) and (900, 1150) for (x_{2}, y_{2}).
m = (1150 - 750)/(900 - 600)
m = 400/300
m = 4/3
Step 4 :
Find the y-intercept.
Use the slope 4/3 and one of the ordered pairs (600, 750).
Slope-intercept form :
y = mx + b
Substitute m = 4/3, x = 600 and y = 750.
750 = (4/3)(600) + b
750 = (4)(200) + b
750 = 800 + b
-50 = b
Step 5 :
Substitute the slope and y-intercept.
Slope-intercept form
y = mx + b
Substitute m = 4/3 and b = -50.
y = (4/3)x + (-50)
y = 4x/3 - 50
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