Solving Equations A Step By Step Guide

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In the realm of mathematics, solving equations is a fundamental skill. Linear equations, in particular, form the bedrock of many mathematical concepts and find widespread applications in various fields, including science, engineering, and economics. This comprehensive guide will delve into the intricacies of solving the linear equation (3/4)x + 3 - 2x = -1/4 + (1/2)x + 5, providing a step-by-step approach that will empower you to tackle similar problems with confidence.

To solve this equation, our primary goal is to isolate the variable x on one side of the equation. This involves a series of algebraic manipulations, each designed to simplify the equation while maintaining its balance. Let's embark on this journey, breaking down each step with clarity and precision.

Step 1: Combining Like Terms

The first step in solving any equation is to simplify both sides by combining like terms. Like terms are terms that contain the same variable raised to the same power. In our equation, we have two terms with x: (3/4)x and -2x on the left side, and (1/2)x on the right side. We also have constant terms: 3 on the left side, and -1/4 and 5 on the right side.

Let's begin by combining the x terms on the left side: (3/4)x - 2x. To do this, we need a common denominator for the fractions. The common denominator for 4 and 1 is 4. So, we can rewrite -2x as (-8/4)x. Now we have (3/4)x - (8/4)x, which simplifies to (-5/4)x. Our equation now looks like this:

(-5/4)x + 3 = -1/4 + (1/2)x + 5

Next, let's combine the constant terms on the right side: -1/4 + 5. To add these, we need a common denominator, which is 4. We can rewrite 5 as 20/4. Now we have -1/4 + 20/4, which simplifies to 19/4. Our equation now becomes:

(-5/4)x + 3 = (1/2)x + 19/4

Step 2: Isolating the Variable Term

Our next objective is to isolate the variable term, which means getting all the x terms on one side of the equation and all the constant terms on the other side. To achieve this, we can add (5/4)x to both sides of the equation. This will eliminate the x term from the left side. Adding (5/4)x to both sides, we get:

3 = (1/2)x + (5/4)x + 19/4

Now, we need to combine the x terms on the right side. To do this, we need a common denominator for 2 and 4, which is 4. We can rewrite (1/2)x as (2/4)x. Now we have (2/4)x + (5/4)x, which simplifies to (7/4)x. Our equation now looks like this:

3 = (7/4)x + 19/4

Next, we need to isolate the x term further by getting rid of the constant term on the right side. To do this, we can subtract 19/4 from both sides of the equation:

3 - 19/4 = (7/4)x

To subtract these, we need a common denominator, which is 4. We can rewrite 3 as 12/4. Now we have 12/4 - 19/4, which simplifies to -7/4. Our equation now becomes:

-7/4 = (7/4)x

Step 3: Solving for x

We're now in the final stretch! To solve for x, we need to get it all by itself. Since x is being multiplied by 7/4, we can divide both sides of the equation by 7/4. Remember that dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of 7/4 is 4/7. So, we can multiply both sides by 4/7:

(-7/4) * (4/7) = (7/4)x * (4/7)

On the left side, the 4s cancel out, and the 7s cancel out, leaving us with -1. On the right side, the 4s cancel out, and the 7s cancel out, leaving us with x. Our equation now simplifies to:

-1 = x

Therefore, the solution to the equation is x = -1. This indicates that if we substitute -1 for x in the original equation, both sides of the equation will be equal.

Step 4: Verifying the Solution (Optional but Recommended)

To ensure our solution is correct, it's always a good practice to verify it by substituting the value of x back into the original equation. Let's substitute x = -1 into our original equation:

(3/4)(-1) + 3 - 2(-1) = -1/4 + (1/2)(-1) + 5

Simplifying the left side, we get:

-3/4 + 3 + 2 = -3/4 + 5

Combining the constant terms, we get:

-3/4 + 5

Simplifying the right side, we get:

-1/4 - 1/2 + 5

To add these, we need a common denominator, which is 4. We can rewrite -1/2 as -2/4 and 5 as 20/4. Now we have:

-1/4 - 2/4 + 20/4

Combining the terms, we get:

17/4

Now, let's go back to the left side and simplify -3/4 + 5. We need a common denominator, which is 4. We can rewrite 5 as 20/4. Now we have:

-3/4 + 20/4

Combining the terms, we get:

17/4

Since both sides of the equation simplify to 17/4, our solution x = -1 is indeed correct. This verification step reinforces the accuracy of our solution and ensures that we have solved the equation successfully.

  • Always simplify both sides of the equation first. This involves combining like terms and clearing any parentheses.
  • Use inverse operations to isolate the variable. To undo addition, use subtraction; to undo subtraction, use addition; to undo multiplication, use division; and to undo division, use multiplication.
  • Perform the same operation on both sides of the equation. This maintains the balance of the equation and ensures that the solution remains valid.
  • Check your solution by substituting it back into the original equation. This verifies the accuracy of your solution.
  • Practice regularly to build your skills. The more you practice, the more comfortable you will become with solving linear equations.

Linear equations are not just abstract mathematical concepts; they have practical applications in various real-world scenarios. Here are a few examples:

  • Calculating costs: Linear equations can be used to calculate the total cost of items or services, such as the cost of a taxi ride or the cost of renting a car.
  • Determining distances: Linear equations can be used to determine distances, such as the distance traveled by a car or the distance between two cities.
  • Converting units: Linear equations can be used to convert units, such as converting Celsius to Fahrenheit or converting kilometers to miles.
  • Solving word problems: Many word problems can be modeled using linear equations, allowing us to find solutions to real-world problems.

Solving linear equations is a fundamental skill in mathematics with wide-ranging applications. By following the steps outlined in this guide, you can confidently solve linear equations and apply them to various real-world scenarios. Remember to practice regularly, and you'll become proficient in solving these equations in no time. Master linear equations and open doors to more advanced mathematical concepts and problem-solving techniques. This guide has walked you through each step to solve the equation and helped you understand the underlying principles. Embrace the challenge and conquer linear equations!

With a solid understanding of how to solve for x, you're well-equipped to tackle more complex mathematical problems. Linear equations are the building blocks for many advanced topics, so mastering them is a crucial step in your mathematical journey. Keep practicing, and you'll find that solving equations becomes second nature. Remember, solving equations is not just about finding the right answer; it's about developing critical thinking and problem-solving skills that will benefit you in all areas of life. So, keep solving and keep learning!