**Solving Word Problems Using Linear Equations :**

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.

**Example 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 = (y₂ - y₁) / (x₂ - x₁)

Substitute (0, 82) for (x₁, y₁) and (1, 80) for (x₂, y₂).

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

Plug m = -2, and (x, y) = (0, 82)

82 = -2(0) + b

82 = 0 + b

82 = b

**Step 4 : **

Now, plug 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.

Plug x = 7 in the equation y = -2x + 82.

y = -2(7) + 82

y = -14 + 82

y = 68

Hence, the temperature at the 7th hour is 68⁰ F.

**Example 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.

Plug 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.

**Example 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 = (y₂ - y₁) / (x₂ - x₁)

Substitute (100, 14) for (x₁, y₁) and (200, 20) for (x₂, y₂).

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

Plug m = 0.06, and (x, y) = (100, 14)

14 = 0.06(100) + b

14 = 6 + b

8 = b

**Step 4 : **

Now, plug 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.

Plug x = 800 in the equation y = 0.06x + 8.

y = 0.06(800) + 8

y = 48 + 8

y = 56

Hence, the cost of plan for 800 minutes included is $56.

**Example 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 = (y₂ - y₁) / (x₂ - x)

Substitute (600, 750) for (x₁, y₁) and (900, 1150) for (x₂, y₂).

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

Plug 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

Plug 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.

Plug x = 1200 in the equation y = (4/3)x - 50.

y = (4/3)(1200) - 50

y = 1600 - 50

y = 1550

Hence, the rent for 1200 square feet of space is $1550.

After having gone through the stuff given above, we hope that the students would have understood, how to solve word problems using linear equations.

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