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.
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.
Problem 3 :
A salesperson receives a weekly salary plus a commission for each computer sold. The table shows the total pay, y, and the number of computers sold, x. Write an equation in slope-intercept form to represent this situation.
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 :
(x_{1}, y_{1}) = (0, 82)
(x_{2}, y_{2}) = (1, 80)
Then,
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 :
(x_{1}, y_{1}) = (100, 14)
(x_{2}, y_{2}) = (200, 20)
Then,
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 :
A salesperson receives a weekly salary plus a commission for each computer sold. The table shows the total pay, y, and the number of computers sold, x. Write an equation in slope-intercept form to represent this situation.
Solution :
Step 1 :
Notice that the change in total pay is the same for increase in sales of every 2 computers. 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 (4, 550) and (6, 700).
Use the slope formula.
m = (y_{2} - y_{1}) / (x_{2} - x_{1})
Substitute :
(x_{1}, y_{1}) = (4, 550)
(x_{2}, y_{2}) = (6, 700)
Then,
m = (700 - 550) / (6 - 4)
m = 150 / 2
m = 75
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 = 75, and (x, y) = (4, 550).
550 = 75(4) + b
550 = 300 + b
250 = b
Step 4 :
Now, substitute m = 75 and b = 250 in slope-intercept form equation of a line.
y = mx + b
y = 75x + 250
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