The document discusses how linear equations are used in everyday situations to model relationships where one variable depends on another. It provides examples of using linear equations to calculate taxi fares based on distance, compare hourly wages, predict costs and profits, and estimate a baby's monthly weight gain. Linear equations allow predicting outcomes by modeling rate of change over time.

1. Linear Function and
Its Application

2. Distance
(in meters)
x
5 km
Lunao
8km
Agay-ayan
11km
Minlagas
13km
San Luis
18km
Medina
23km
Amount
(in Php)
y
Real-life problems involving linear functions.
Going to a far barangay in Gingoog City means riding a
multicab. The standard fare in riding a multicab is Php 7 for the
first 5 km as a flag down rate plus Php 1 peso for every 1
kilometer. How much is Nathan going to pay if he goes to
Medina?
Complete the table:
Distance
(in meters)
x
5 km
Lunao
8km
Agay-ayan
11km
Minlagas
13km
San Luis
18km
Medina
23km
Amount
(in Php)
y

3. Multicab Ride!

4. How Are Linear Equations Used in Everyday Life?
Linear equations use one or more variables where one variable is
dependent on the other. Almost any situation where an unknown quantity can be
represented by a linear equation, like figuring out income over time, calculating
mileage rates, or predicting profit. Many people use linear equations every day,
even if they do the calculations in their head without drawing a line graph.
o Variable Costs
Imagine that you are taking a taxi while on vacation. You know that the taxi service
charges 100 pesos to pick your family up from your hotel and another 5 pesos per kilometer
for the trip. Without knowing how many kilometers it will be to each destination, you can
set up a linear equation that can be used to find the cost of any taxi trip you take on your
trip. By using "x" to represent the number of kilometers to your destination and "y" to
represent the cost of that taxi ride, the linear equation would be: y = 5x + 100.

5. o Rates
Linear equations can be a useful tool for comparing rates of pay.
For example, if one company offers to pay you 450 pesos per week and the other offers 10 pesos
per hour, and both ask you to work 40 hours per week, which company is offering the better rate of
pay? A linear equation can help you figure it out! The first company's offer is expressed as 450 =
40x. The second company's offer is expressed as y = 10(40). After comparing the two offers, the
equations tells you that the first company is offering the better rate of pay at 11.25 pesos per hour.
o Budgeting
A party planner has a limited budget for an upcoming event. She'll need to figure out how
much it will cost her client to rent a space and pay per person for meals. If the cost of the rental
space is 780 pesos and the price per person for food is 100 per person, a linear equation can be
constructed to show the total cost, expressed as y, for any number of people in attendance, or x.
The linear equation would be written as y = 100x + 780. With this equation, the party planner can
substitute any number of party guests and give her client the actual cost of the event with the
food and rental costs included.

6. o Making Predictions
One of the most helpful ways to apply linear equations in everyday life is to make
predictions about what will happen in the future.
If a bake sale committee spends 200 pesos in initial start-up costs and then earns 150
pesos per month in sales, the linear equation y = 150x - 200 can be used to predict cumulative
profits from month to month. For instance, after six months, the committee can expect to have
netted 700 pesos because (150 x 6) - 200 = 700. While real world factors certainly impact how
accurate predictions are, they can be a good indication of what to expect in the future. Linear
equations are a tool that make this possible.

7. My Baby Bro!
You have a newly-born baby brother. Suppose the baby weights 3kg at birth. You’ve
known from your mother that the monthly average weight gained by the baby is 1kg. Suppose
the rate of increase in the baby’s weight every month is constant, determine an equation that
will describe the baby’s weight. Predict the baby’s weight after five months using mathematical
equation and graphical representation.
Follow the flowchart below then use it to answer the questions that follow:

8. Rate of Change: Real World Application
Sample Problem # 1 Marathon Athlete
An athlete begins the normal practice for the next marathon during the evening. At 6:00 pm he starts to
run and leaves his home. At 7:30 pm, the athlete finishes the run at home and has run a total of 7.5
miles. How fast was his average speed over the course of the run?
Solution:
 The rate of change is the speed of his run, distance over time. Therefore, the two variables are time
(X) and distance (Y).
 The first point is at his house, where his watch read 6:00 pm. This is the beginning time so let’s set it
to 0. So, our first point is (0,0) because he did not run anywhere yet.
 Let’s think about our time in hours. Our second point is 1.5 hours later, and we ran 7.5 miles. The
second point is (1.5,7.5). Our speed (rate of change) is simply the slope of the line connecting the two
points.
 The slope given by: m =
𝑦2−𝑦1
𝑥2−𝑥1
becomes m =
7.5 𝑚𝑖𝑙𝑒𝑠
1.5 ℎ𝑜𝑢𝑟
= 5 miles per hour.

9. Sample Problem # 2 Renting a Moving Van
A rental company charges a flat fee of ₱300 and an additional ₱25 per mile to rent a moving van. Write a
linear equation to approximate the cost Y (in pesos) in terms of X, the number of miles driven. How
much would a 75 mile trip cost?
Solution:
 Using the slope-intercept form of linear equation, with the total cost labeled Y (dependent variable)
and the miles labeled X (independent variable):
𝑦 = 𝑚𝑥 + 𝑏
 The total cost is equal to the rate per mile times the number of miles driven plus the cost for the flat
fee: 𝑦 = 25𝑥 + 300
 To calculate the cost of a 75-mile trip, substitute 75 for x into the equation:
𝑦 = 25𝑥 + 300
𝑦 = 25(75) + 300
𝑦 = 1,875 + 300
𝑦 = 2,175 pesos

10. Choose the letter of the correct answer.
1. What is abscissa.
a. It is an x-coordinate.
b. It is a y-coordinate.
c. It divides the plane into four regions called quadrant.
d. It is a point on the xy-plane.
2. Which best describes the point (5, -2)?
a. It is two units above the x-axis and five units to the left of the y-axis.
b. It is two units above the x-axis and five units to the right of the y-axis.
c. It is two units below the x-axis and five units to the left of the y-axis.
d. It is two units below the x-axis and five units to the right of the y-axis.
For items 3 to 5, refer to the situation below:
3. Which of the following equations best represents the total cost y
with x number of square feet including lay outing fee?
a. y = 150x – 15 c. y = 15x + 150
b. y = 150x + 15 d. y = 15x – 150
4. What qualities you must look into in tarpaulin printing?
I. The quality of the layout artist’s output
II. The brand of the PC used in lay outing
III. The quality of the printing output
IV. The printing and lay outing cost
a. I and II only c. I, III, and IV only
b. I, II and III only d. II, III and IV only
Janrey, who is the school office assistant, was given the task by the School Principal
to canvass for a tarpaulin printing for the opening of classes. He knew that in printing
ad, the charge of tarpaulin is Php 15 per square foot and Php 150 for lay outing.
5. The School Principal told Janrey that the dimensions of the
tarpaulin are 8 feet by 6 feet. How many square feet is the
tarpaulin? How much should Janrey pay for the printing ad?
a. 48 square feet; Php 720 c. 14 square feet; Php 210
b. 48 square feet; Php 870 d. 14 square feet; Php 310
6. What will happen to the value of y in the equation 4x + 6y = 10
when the value of x decreases?
a. The value of y will not change. c. The value of y will increase.
b. The value of y cannot be determined. d. The value of y will decrease.
7. Marlon rode a multicab from a bus terminal to San Luis National High
School, whose distance is approximately 15 kilometers. After riding, he
paid an amount of Php 20. Which variable is dependent?
a. The amount paid c. The person riding the multicab
b. Multicab riding d. The distance traveled
For items 8 and 9, refer to the situation:
9. If Hon. Cagumbay has to choose one best representation of the
relationship between electric bill and power consumption in his power
point presentation, what do you think he should use to present his ideas in
the clearest way?
a. Graph b. table c. mapping diagram d. rule of equation
10. A car travels at a uniform speed. It covers a distance of 60km in an
hour, a distance of 120km in 2 hours, and 180km in 3hours. How far can it
travel in 4 hours?
a.240km b. 210km c. 260km d. 280km
In a particular barangay, you are elected as the Barangay Chairman. Hon.
Cagumbay, who is a councilor, was assigned as the chairman of Committee on
Energy. You gave him a task to make a Power point presentation illustrating the
relationship between the electric bill and power consumption, and provide
recommendations and friendly reminders to minimize energy consumption.