**Solving word problems with linear equations worksheet :**

Worksheet on solving word problems with linear equations is much useful to the students who would like to practice solving real-world problems in algebra.

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.

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 ?

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.

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.

**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 = (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.

Let us look at the next problem on "Solving word problems with linear equations worksheet"

**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.

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.

Let us look at the next problem on "Solving word problems with linear equations worksheet"

**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 = (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.

Let us look at the next problem on "Solving word problems with linear equations worksheet"

**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 = (y₂ - y₁) / (x₂ - x₁)

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.

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