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Solving Linear Equations In One And Two Variables, Including Word Problems

linear equations

Linear equations are like the sturdy building blocks of mathematics, forming the foundation upon which countless mathematical principles and real-world applications are constructed. At first glance, they may appear somewhat intimidating, but rest assured, once you unravel their core concepts, linear equations become far more approachable. In this straightforward guide, we’ll demystify the world of linear equations, demonstrating how to solve them in one and linear equations in two variables. We’ll even toss in some word problems to further hone your skills.

Understanding Linear Equations

It’s important to guarantee a solid understanding of what a linear equation is before we enter into the world of solving linear equations. In simple terms, a linear equation is one in which the variables are increased to a power of 1 or less. In simpler terms, you won’t come across variables squared, cubed, or raised in any other way within a linear equation.

One-Variable Linear Equations

Let’s embark on our journey by delving into the fundamental concepts: one-variable linear equations. These equations centre around a lone variable often symbolised as ‘x.’ Here’s a quintessential example of a one-variable linear equation:

5x + 3 = 18

Your objective? To unearth the value of ‘x’ that harmonises with this equation. The key to solving this equation lies in the art of isolating ‘x’ on one side of the equation, step by step:

  1. Subtract 3 from both sides to isolate the term with ‘x’:

    5x = 18 – 3

  1. Streamline the equation’s correct side:

    5x = 15

  1. Now, let’s divide both sides by 5 to pinpoint the value of ‘x’:

    x = 15 / 5

  1. Further: simplify the equation:
  2. x = 3

You’ve successfully unravelled the mystery of this one-variable linear equation. In this particular case, ‘x’ shines bright at the value of 3.

Also Read: Similarity of Triangles: Examining criteria for similarity and solving problems related to similar triangles.

Two-Variable Linear Equations

Now, let’s advance to linear equations in two variables. These equations introduce two variables, often symbolised as ‘x’ and ‘y.’ A typical two-variable linear equation might present itself as follows:

3x + 2y = 10

Solving these linear equations in two variables necessitates identifying values for both ‘x’ and ‘y’ that cohesively satisfy the equation. Let’s break it down into a methodical process:

  1. Initiate by isolating one of the variables. In this example, we shall focus on ‘x’ by subtracting ‘2y’ from both sides:

    3x = 10 – 2y

  1. Simplify the equation:

    3x = 10 – 2y

  1. Now, proceed to divide both sides by 3:
  2. x = (10 – 2y) / 3

To obtain values for both variables, you’ll need more information because this equation comprises both ‘x’ and ‘y,’ therefore it’s vital to keep that in mind. However, at this point, you have effectively learned how to isolate a single variable in a linear equation with two variables.

Also Read: Constructions: Learning to construct geometric figures like bisectors, perpendiculars, and triangles

Word Problems: Putting Linear Equations into Action

Linear equations are invaluable tools when it comes to unravelling real-world dilemmas. Let’s delve into some practical word problems where you can apply your burgeoning expertise in solving linear equations.

Word Problem 1

Imagine you’re gearing up for an exciting road trip, and your trusty car boasts a fuel efficiency rating of 25 miles per gallon. With a full tank of gas, accommodating 15 gallons, how far can you venture before your gas gauge nudges the ‘E’?

To conquer this problem, we can set up a one-variable linear equation. Let ‘m’ embody the number of miles you can journey, while ‘g’ signifies the number of gallons of fuel you exhaust:

m = 25g

With this equation at your disposal, let’s substitute the values:

m = 25 * 15

m = 375

Eureka! You can savour 375 miles of scenic road before your next pit stop for fuel.

Word Problem 2

Suppose you’re on a school supply shopping spree, and you’ve already allocated $30 for notebooks and pencils. A budget of $70 remains to be spent on textbooks, each priced at $15. How many textbooks can you add to your academic arsenal?

Let ‘t’ symbolise the number of textbooks you can proudly possess:

15t + 30 = 70

    Now, it’s time to unearth the value of ‘t’:

  1. Commence by subtracting 30 from both sides to isolate ’15t’:

    15t = 70 – 30

  1. Streamline the equation:

    15t = 40

  1. Finally, execute the division operation on both sides by 15 to unlock the number of textbooks:
  2. t = 40 / 15

    t = 2.67

Since textbooks come in whole units, you can purchase 2 textbooks with your $70 budget.

Word Problem 3

Visualise hosting a birthday celebration replete with delicious pizza. Each pizza is divided into 8 equally scrumptious slices, and your 12 guests are anticipated to relish 3 slices each. The question at hand: How many pizzas should you order to ensure that everyone’s pizza cravings are satisfied?

Let ‘p’ signify the quantity of pizzas you should order to orchestrate a pizza extravaganza:

8p = 12 into 3

Let’s compute the solution:

8p = 36

p = 36 / 8

p = 4.5

However, given the undeniable truth that you can’t procure half a pizza, it’s prudent to round up your order to 5 pizzas, guaranteeing that all your guests have ample pizza to savour.

Also Read: Different types of quadrilaterals: Parallelograms And Rectangles

Linear equations are the cornerstone of mathematical problem-solving, guiding you toward clarity and informed decision-making. Whether you’re tackling one-variable or linear equations in two variables, the core approach remains consistent: isolate the variable and find the solution. These equations provide a structured path to unveiling answers in the world of mathematics.

Word problems are real-world scenarios that test your prowess in solving linear equations. Breaking these scenarios down into equations grants you a direct method for uncovering solutions. Armed with these skills, you’re ready to confidently face mathematical challenges and navigate real-life dilemmas. Linear equations become your reliable allies in the realm of numbers and logic, ensuring you find clear and precise solutions. Embrace the journey of problem-solving with newfound confidence, knowing that linear equations are there to support you every step of the way. Happy problem-solving!

At EuroSchool, we place a strong emphasis on equipping students with essential mathematical skills, including proficiency in solving linear equations. Our comprehensive curriculum covers the art of handling linear equations in one and two variables, preparing students to excel in the world of mathematics. Through real-world applications and engaging word problems, we challenge our students to think critically and apply their knowledge to practical situations.

By mastering linear equations, our students gain not only mathematical expertise but also problem-solving abilities that serve them well in their academic journey and beyond. At EuroSchool, we are committed to nurturing a deep understanding of linear equations and empowering students to confidently tackle any mathematical challenge that comes their way.

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