Linear Functions Word Problems Practice & Solutions

Master representing real-world phenomena with linear functions through engaging practice problems. Solve truck movement, rate problems, and more with step-by-step solutions.

📚What You'll Master in This Practice Session
  • Create linear equations from real-world scenarios like truck movement problems
  • Interpret intersection points to find when two moving objects meet
  • Calculate rates of change and slopes from linear function graphs
  • Solve systems of linear equations using graphical methods
  • Convert word problems into mathematical representations using coordinates
  • Analyze distance-time relationships using linear function properties

Understanding Representation of Phenomena

Complete explanation with examples

A linear function describes the relationship between X X and Y Y .
Therefore, we can represent all sorts of different phenomena in life with the help of the linear function.
The representation of phenomena with the help of linear functions is expressed in mathematics in word problems, using graphs of the functions.
Thus, we can find the various relationships between the functions.

Representing phenomena using linear functions actually allows us to simplify many word questions using a simple linear graph. From the graph, we can very easily calculate the slope, which is actually the rate of change and even many other parameters.

A - Representation of Phenomena Using Linear Functions

Detailed explanation

Practice Representation of Phenomena

Test your knowledge with 2 quizzes

Two cyclists go out for a ride.

The first one starts at 4, while the second starts at 5.

At what times do the riders take a break?

Time000111222333444555666777888999101010111111121212131313141414151515161616555101010151515202020252525303030353535404040454545111222

Examples with solutions for Representation of Phenomena

Step-by-step solutions included
Exercise #1

The graph below depicts the price of apples relative to the quantity.

What is the price of 7 kg of apples?

–3–3–3–2–2–2–1–1–1111222333444555666777888999101010111111121212131313141414151515161616171717181818191919202020111222333444555666777888999101010111111121212kg de manzanasPrecio

Step-by-Step Solution

Let's first observe the X-axis which is representative of the weight. Locate 7 kg - and mark a point.

From this point, we'll draw a vertical line until we meet the straight line.

Let's now observe the Y-axis which is representative of the price.

We will draw a line from the point where we reached the X-axis to the straight line, towards the Y-axis and determine the price: 8 dollars.

Answer:

8

Video Solution
Exercise #2

The graph below depicts the price of apples relative to the quantity.

How many kg of apples can you buy for $9?

–3–3–3–2–2–2–1–1–1111222333444555666777888999101010111111121212131313141414151515161616171717181818191919202020111222333444555666777888999101010111111121212kg de manzanasPrecio

Step-by-Step Solution

Let's first observe the Y-axis which is representative of the price. Locate 9 dollars - and mark a point.

From this point, we'll draw a vertical line until we meet the straight line.

Now let's observe the X-axis which is representative of the quantity in kg.

We'll draw a line from the point where we reached the Y-axis to the straight line, towards the X-axis and determine the weight: 8 kg.

Answer:

8

Video Solution
Exercise #3

The graph below depicts the price of tomatoes as a function of quantity, with each line representing the prices at different shops.

Where is it cheaper to buy 7 kg of tomatoes?

–3–3–3–2–2–2–1–1–1111222333444555666777888999101010111111121212131313141414151515161616171717181818191919–1–1–1111222333444555666777888999101010111111kg de tomatesPrecio

Step-by-Step Solution

Let's look at the red line:

The X-axis shows us the quantity, we'll locate 7 kg and from it draw a vertical line to the line - mark a point.

From this point we'll draw a horizontal line to the Y-axis which shows the price and we'll see that 7 kg costs 8 NIS.

Let's look at the blue line:

The X-axis shows us the quantity, we'll locate 7 kg and from it draw a vertical line to the line - mark a point.

From this point we'll draw a horizontal line to the Y-axis which shows the price and we'll see that 7 kg costs 10 NIS.

Therefore it's better to buy at the red store, it's cheaper.

Answer:

In the red shop

Video Solution
Exercise #4

There are two water containers. The first container is empty, whilst the second contains 48 liters of water.

How long will in take for the second container to reach full capacity?

Minutos000222444666888101010121212141414161616181818202020222222242424262626282828Litros484848565656646464727272808080888888969696104104104112112112120120120128128128136136136144144144152152152160160160AAABBB

Step-by-Step Solution

Let's begin by observing the blue line that represents the second container.

Note that it will be full at the point where it starts to become a continuous line.

Hence let's draw a vertical line from this point to the X-axis which marks the minutes.

We can observe that the line reaches 22, meaning the container will be full in 22 minutes.

Answer:

22 minutes

Exercise #5

Two cyclists go for a ride along the same path.

The first cyclist leaves at 4, while the second cyclist leaves at 5.

At what time do they meet?

Time000111222333444555666777888999101010111111121212131313141414151515161616555101010151515202020252525303030353535404040454545111222

Step-by-Step Solution

The problem asks us to determine if the two cyclists meet after departing at different times. To solve this, we would analyze the functions or the possible graphical representation of their journey concerning time.

From the provided scenario, cyclist one starts at 4 and cyclist two at 5. Without specific speed and distance, the problem might hint at a graphical or conceptual analysis.

Upon examining this scenario, since no intersection of their paths was indicated (or given speeds and times to compute), we need to conclude based on the apparent description or plot.

Without specific data points indicating overlap or meeting times between the two cyclists, we assume the visual information presented suggests that they indeed do not encounter each other on their paths.

Therefore, the cyclists do not meet, confirming the conclusion as per the given choices.

Therefore, the solution to the problem is: The cyclists do not meet.

Answer:

The cyclists do not meet.

Frequently Asked Questions

Everything you need to know about Representation of Phenomena

How do you represent real-world phenomena using linear functions?

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Linear functions represent real-world phenomena by establishing relationships between two variables, typically expressed as y = mx + b. For example, in truck movement problems, time becomes the x-variable and distance from a reference point becomes the y-variable, allowing us to track position changes over time.

What does the intersection point mean in linear function word problems?

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The intersection point represents when two phenomena meet or have equal values. In truck problems, it shows exactly when and where two vehicles traveling toward each other will meet, providing both the time elapsed and the distance from the starting reference point.

How do you find the equation of a line from two points in word problems?

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To find the equation from two points: 1) Calculate the slope using m = (y₂-y₁)/(x₂-x₁), 2) Use point-slope form: y - y₁ = m(x - x₁), 3) Simplify to y = mx + b format. This method works for any linear relationship in real-world scenarios.

What are common mistakes when solving linear function word problems?

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Common mistakes include: misidentifying which variable represents time vs. distance, incorrectly interpreting starting positions from graphs, forgetting to check if the intersection point makes sense in the real-world context, and confusing positive vs. negative slopes for approaching vs. departing movements.

How do you interpret the slope in distance-time linear function problems?

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The slope represents the rate of change, which is speed in distance-time problems. A positive slope means moving away from the reference point, while a negative slope means moving toward it. The steeper the slope, the faster the speed of movement.

Why are linear functions useful for representing phenomena in mathematics?

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Linear functions simplify complex real-world relationships into manageable mathematical models. They allow us to predict future values, find intersection points, calculate rates of change, and solve problems graphically or algebraically using familiar techniques like systems of equations.

What information do you need to create a linear function from a word problem?

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You need: 1) Two distinct points or one point plus the rate of change, 2) Clear identification of independent (usually time) and dependent variables, 3) Understanding of the starting conditions, 4) The direction of change (increasing or decreasing relationship).

How do you solve truck meeting problems using linear functions?

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Steps to solve truck meeting problems: 1) Set up coordinate system with time as x-axis and distance from reference point as y-axis, 2) Write linear equations for each truck using given points, 3) Set equations equal to find intersection, 4) Solve for time and substitute back to find meeting distance.

Continue Your Math Journey

Master Representation of Phenomena first, then advance to these related topics that build on your fraction skills

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