We can write a linear equation for the information found in the given real-world problem and solve the problem using the linear equation.
In most of the cases, we use slope-intercept form equation to solve the real-world problems.
Problem 1 :
The table shows the temperature of a fish tank during an experiment. Write the appropriate linear equation for the given situation and use the equation to find temperature at the 7th hour.
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.
Since we want to find the temperature at the 7th hour, the appropriate linear equation for the given situation is slope-intercept form (y = mx + b), assuming y as temperature and x as hours.
Step 2 :
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
Step 5 :
Find the temperature at the 7th hour.
Substitute x = 7 in the equation y = -2x + 82.
y = -2(7) + 82
y = -14 + 82
y = 68
So, the temperature at the 7th hour is 68⁰ F.
Problem 2 :
Lily has just opened her new computer store. She makes $25 on every computer she sells and her monthly expenses are $10,000. What is the minimum number of computer does she need to sell in a month to make a profit ?
Solution :
Step 1 :
Let y stand for the profit and x stand for number of computers sold.
From the given information, we have
Profit = 27 x No. of computers sold - Monthly expenses
y = 27x -10,000
Step 2 :
Let us find the number of computers sold for no profit.
That is, find the value of x when y = 0.
Substitute y = 0 in the equation y = 27x - 10,000.
0 = 25x - 10,000
Add 10,000 to both sides.
10,000 = 25x
Divide both sides by 25.
10,000 / 25 = 25x / 25
400 = x
Step 3 :
When Lily sells 400 computers in a month, her profit is equal to zero.
So, she has to sell more than 400 computer per month to make a profit.
To make a profit, the minimum number of computers per month, she needs to sell is 401.
Problem 3 :
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 and use it to estimate cost of plan for 800 minutes included.
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
Step 5 :
Estimate cost of plan for 800 minutes included.
Substitute x = 800 in the equation y = 0.06x + 8.
y = 0.06(800) + 8
y = 48 + 8
y = 56
So, the cost of plan for 800 minutes included is $56.
Problem 4 :
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 and use it to calculate the rent for 1200 square feet of space.
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 = (4/3)x - 50
Step 6 :
Calculate the rent for 1200 square feet of space.
Substitute x = 1200 in the equation y = (4/3)x - 50.
y = (4/3)(1200) - 50
y = 1600 - 50
y = 1550
So, the rent for 1200 square feet of space is $1550.
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