How Linear Equations Simplify Everyday Maths Problems
There are a number of applications of linear equations in Mathematics and in real life. An algebraic expression consists of variables that are equated to each other using an equal “=” sign, it is called an equation. An equation with a degree of one is termed a linear equation. Mathematical knowledge is usually applied through word problems, and the applications of linear equations are observed on a wide scale to solve such word problems. Here is a detailed discussion of the applications of linear equations and how they will fit in the real world.
Linear Equation
A Linear equation is an algebraic expression that consists of variable and equality sign (=). Linear equations are classified based on the number of variables.
Linear Equation with One Variable
The equation which has one degree and one variable is called a linear equation with one variable.
For Example: 5x +30 = 0, 3x + 12 = 48
Linear Equation with Two Variable
The equation which has one and two variables is called a linear equation with two variables.
For Example: 2x + 3y =12, 3x + 7y = 42
Representation of Linear Equations
The graphical representation of the linear equation is ax + by + c = 0. Where a and b are coefficients, x and y are variables, and c is a constant term.
Applications of Linear Equations
The applications of linear equations are vast and are applicable in numerous real-life situations. To handle real-life situations using algebra, we change the given situation into mathematical statements. So that it clearly illustrates the relationship between the unknown variables and the known information. The following are the steps involved to reiterate a situation into a mathematical statement,
Convert the real problem into a mathematical statement and frame it in the form of an algebraic expression that clearly defines the problem situation.
Identify the unknowns in the situation and assign variables of these unknown quantities.
Read the situation clearly a number of times and cite the data, phrases, and keywords. Sequentially organize the obtained information.
Write an equation using the algebraic expression and the provided data in the statement and solve it using systematic equation solving techniques
Reframe the solution to the problem statement and analyze if it exactly suits the problem.
Using these steps, the applications of word problems can be solved easily.
Applications of Linear Equations in Real life
The following are some of the examples in which applications of linear equations are used in real life.
It can be used to solve age related problems.
It is used to calculate speed, distance and time of a moving object.
Geometry related problems can be solved.
It is used to calculate money and percentage related problems.
Work, time and wages problems can be solved.
Problems based on force and pressure can be solved.
FAQs on Real-Life Applications of Linear Equations
1. What is a linear equation, and why are its applications so common in real life?
A linear equation is an algebraic equation that forms a straight line when plotted on a graph. The standard form is ax + by + c = 0, where x and y are variables. Its applications are extremely common because many real-world relationships involve a constant rate of change. For instance, if you get paid an hourly wage, your total earnings increase at a steady rate for every hour you work. This simple, direct relationship makes linear equations a powerful tool for modelling predictable scenarios.
2. What are some specific examples of how linear equations are applied in everyday situations?
Linear equations are used constantly to model real-world scenarios. Here are a few common examples:
- Calculating Costs: Determining the total cost of a mobile phone plan that has a fixed monthly fee plus a charge per GB of data used.
- Speed, Distance, and Time: Calculating the distance a car will travel in a certain amount of time if it maintains a constant speed.
- Temperature Conversion: Converting temperatures between Celsius and Fahrenheit using the linear formula F = (9/5)C + 32.
- Budgeting: Planning personal or business expenses where a total budget is allocated across different items with fixed costs.
- Age Problems: Solving word problems that involve the ages of different people now, in the past, or in the future.
3. How do you translate a real-world word problem into a mathematical linear equation?
Translating a word problem into a linear equation involves a systematic process:
- Identify the Unknowns: First, determine what quantity you need to find and assign a variable to it (e.g., let 'x' be the number of hours).
- Extract Key Information: Read the problem carefully to find numbers, rates, and totals. Look for keywords like 'is', 'equals', 'per', 'more than', and 'less than'.
- Form the Equation: Arrange the variables and numbers into a logical mathematical statement that represents the situation described in the problem. For example, a statement like "the total cost is a $50 setup fee plus $10 per hour" becomes the equation C = 10x + 50.
- Solve and Verify: Solve the equation for the unknown variable and then check if the answer makes sense in the original context of the problem.
4. Why are concepts like speed, distance, and time often modelled using linear equations?
The relationship between speed, distance, and time is fundamentally linear when the speed is constant. The formula Distance = Speed × Time is a perfect linear relationship. If you plot distance travelled against time, you get a straight line. The slope of this line is the speed. This predictability is why linear equations are the primary tool for solving problems involving objects moving at a steady velocity, allowing us to calculate travel time, distance covered, or the speed required to reach a destination.
5. In business and economics, how are linear equations used for practical calculations?
In business and economics, linear equations are essential for analysis and decision-making. They are used to calculate the break-even point, which is the point where total revenue equals total costs. For example, if a company's cost is C = 1000 + 5x and its revenue is R = 10x, setting C = R helps find the number of units 'x' they must sell to not make a loss. They are also used to model simple supply and demand curves and calculate profit or loss.
6. How does the application of a linear equation in one variable differ from one in two variables?
The difference lies in what they help you find.
- A linear equation in one variable (e.g., 3x + 9 = 21) is used to find a single, specific unknown value. For instance, it can help you find the exact price of one item when you know the total cost and other fixed charges.
- A linear equation in two variables (e.g., y = 4x + 2) is used to describe a relationship between two quantities. It doesn't give you one single answer but a range of possible solutions that lie on a straight line. For example, it can model how the total cost (y) changes based on the number of items purchased (x).
7. What is the importance of the 'slope' and 'y-intercept' in a real-world application of a linear equation?
In the context of a real-world problem modelled by the equation y = mx + c, the components have very specific meanings:
- The slope (m) represents the rate of change. For example, in a cost function, it's the price per item. In a distance-time graph, it's the speed. It tells you how much the 'y' value changes for every one-unit increase in the 'x' value.
- The y-intercept (c) represents the initial or fixed value. It is the value of 'y' when 'x' is zero. For example, it could be a one-time setup fee for a service or the starting distance of a journey from a reference point.
8. Are there any real-world situations where a linear equation is a poor approximation?
Yes, absolutely. Linear equations are a poor fit for situations where the rate of change is not constant. For example, they cannot accurately model:
- Population Growth: Populations often grow exponentially, not in a straight line.
- Compound Interest: The interest earned also earns interest, leading to accelerating growth that is better described by an exponential equation.
- Projectile Motion: The path of a thrown ball is a parabola, which is a quadratic relationship, not a linear one.
In these cases, using a linear model would lead to highly inaccurate predictions.
