A linear equation is a fundamental concept in mathematics that has a wide range of applications in the real world. Straight-line equations are the most common use. The terms, slopes, intercepts, points, and others, are used to describe linear equations.
Linear equations have a surprising number of applications in our daily lives. We will look at some of the applications of linear systems in our everyday lives with the help of this blog. But, before we get into the applications of linear systems, let’s define linear equations and some of the terms associated with them.
What are Linear Equations?
Linear equations refer to first-order equations. These equations form a straight line, and a linear equation is represented by the equation y=mx+b, where m denotes the slope. You can use one or more variables in linear equations. A one-variable linear equation is referred to as a linear equation with one variable. You can use a linear equation to depict almost any circumstance involving an unknown number, such as estimating income over time, computing mileage rates, or predicting profit. Many people use linear equations on a daily basis, even if they don’t visualize a line graph in their heads.
Key terms in linear equations:
- Change in Rate
The rate of change is frequently included in linear equations. Velocity, for example, is the rate of distance variation over time. The slope is a rate of change that could be deduced if we know the total distance that is traveled and the two points in time. Then, the linear equation could be created using this data, and predictions could be made using the linear equation.
If the amount or unit in which something changes is not given, the rate is usually expressed in terms of time. “Per unit of time” rates, such as heart rate, speed, and flux, are the most prevalent. Exchange rates, electric fields, and literacy rates are examples of non-time denominator ratios.
Imagine a roof or a ski slope while thinking about the slope of a line. Roofs and ski slopes can be either steep or relatively flat. In truth, much like lines, ski slopes and roofs can be flat (horizontal). An utterly vertical ski slope or roof would be impossible to find, but a line might.
We can typically tell which ski slope is steeper by looking at it. The three ski slopes get steeper with time.
- Independent Variable
An independent variable is a variable that exists independently of the equation and serves as its input. For instance, if you wanted to see how much water a plant needs to survive, you could test different amounts of water on plants kept in the same lighting and soil conditions. In this case, you’re utilizing water as the independent variable (or input). The letter x denotes the independent variable in a linear equation.
- Dependent Variable
The output, or dependent variable, is the result of the independent variable. For example, after you’ve watered your plants, you might wish to keep track of how much each one has grown. The amount of water you give a plant determines how much it grows. The letter y denotes the dependent variable in a linear equation.
Steps to Solve a Linear Equation:
- Read the Problem Statement
To comprehend what is offered, what type of real-world example of linear function it is, and what is to be found, you must read the problem attentively. Then, if necessary, read it as many times as necessary.
- Allocate Variables
Represent one of the known values or quantities with a variable and use diagrams or tables to tie all of the other unknown values (if any) to this variable. Make a list of what each variable stands for.
- Write the Equations
Create equations that connect unknown and known quantities. We can use some of the well-known formulas and the figure/equations outlined in the preceding phase to find the applicable equation that will lead to the result we want.
- Solve the Equation
Solve the equations you created in the previous stage and answer all of the questions because the equation will only give you one of the values you asked for.
- Examine the Solutions
You can confirm the solution by entering it into the equation, but make sure it’s correct.
Let’s look at some of the linear function’s real-life examples now that we know what they are and how they work.
Linear Equations in Practice
You might be shocked to learn that linear equations have vital applications in our daily lives in various industries. We’ll look at some of the real-life examples of linear functions in this section:
Using linear equations, you can estimate the expenses and charges of various items without any missing quantities. For example, let’s say you’re trying to figure out how much a cab will cost, and you don’t know how far you’ll be traveling. Assuming x represents the distance traveled, you can rapidly form a linear equation. The math becomes simple in this manner.
Assume you’re on vacation and need to take a taxi. You’re aware that the taxi service will charge $9 to pick up your family from your hotel, plus $0.15 for every mile after that. You can use a linear equation to determine the cost of whatever cab trip you take on your vacation without knowing how many miles it will be to each location. For example, the linear equation would be y = 0.15x + 9 if “x” represents the number of miles to your destination and “y” represents the cost of that taxi fare.
When it comes to budgeting, a lot of individuals use linear equations. Likewise, many large corporations use linear equations to estimate their budgets and product costs. Budgeting with linear equations allows these businesses to provide better prices to their customers, allowing them to compete successfully.
A party planner has a limited budget for an upcoming event. She’ll have to calculate how much it will cost her customer to hire a location and pay for meals per participant. You may write a linear equation to illustrate the total cost, expressed as y, for any number of people in attendance, or x if the rental space is $780 and food costs $9.75 per person. In this case, the linear equation would be y = 9.75x + 780. The party planner can use this equation to substitute any number of party participants and tell her client the total cost of the event, including food and rental costs.
Linear equations are an excellent tool for comparing rates. For example, let’s say two companies offer you x dollars for y hours of work. Using linear equations, you may choose which of these organizations offers you a better rate for the number of hours you work. One of the most common uses of linear equations is in this situation.
When comparing salary rates, linear equations can be a valuable tool. For example, if one company provides $450 per week and the other offers $10 per hour, both companies require you to work 40 hours per week. Which one is the better deal? You can use a linear equation to figure it out! The first firm’s offer is calculated as 450 = 40x. The second firm’s offer is written as y = 10. (40). After comparing the two offers, the calculations show that the first company pays $11.25 per hour, which is better.
This is unexpected but true! Daily, linear equations assist in formulating numerous forecasts. For example, many start-ups employ linear equations to forecast how they will perform in the future and the cumulative profits for each month. Although many real-life examples of linear functions are considered when forecasting, linear equations come in handy in these situations.
Making predictions about what the future will look like is one of the most useful ways to use linear equations in everyday life. The linear equation y = 150x − 200 can estimate cumulative profits from month to month if a bake sale committee pays $200 in initial start-up expenditures and subsequently earns $150 per month in sales. For example, the committee can expect to have earned $700 after six months since (150 x 6) − 200 = $700. While linear functions in real-life events undoubtedly influence the accuracy of projections, they can provide a useful signal of what to expect in the future. This is possible through the use of linear equations.
We don’t like learning about linear systems or linear functions in school because we don’t understand or see how they relate in real life. However, as a business and economics application of linear systems, as well as real-life examples of linear functions, these concepts serve a useful tool for navigating and finding solutions. The trick is to figure out which linear formula or concept may be applied to linear functions in real life.