Solving Linear Equations Step-by-Step A Comprehensive Guide

by James Vasile 60 views

Hey guys! Ever get tangled up in linear equations? Don't worry, it happens to the best of us. Linear equations might seem intimidating at first, but with a little know-how, they're totally solvable. In this article, we're going to break down a specific linear equation step-by-step, so you can conquer similar problems with confidence. We'll focus on this equation:

d−10−2d+7=8+d−10−3dd - 10 - 2d + 7 = 8 + d - 10 - 3d

And we'll figure out which of these options is the correct solution:

A. $d = 5$ B. $d = -1$ C. $d = -5$ D. $d = 1$

So, grab your thinking caps, and let's dive in!

Understanding Linear Equations

Before we jump into solving our equation, let's quickly recap what linear equations are all about. A linear equation is basically a mathematical statement that shows the equality between two expressions. These expressions involve variables (like our 'd' here) raised to the power of 1 (that's what makes them linear). There are no exponents, square roots, or other fancy operations on the variable.

The goal when solving a linear equation is to isolate the variable on one side of the equation. This means we want to get something like "d = some number". To do this, we use algebraic manipulations, which are just operations that maintain the equality. We can add, subtract, multiply, or divide both sides of the equation by the same value without changing the solution. It's like a balancing act – whatever you do to one side, you have to do to the other!

Why are linear equations important? Well, they pop up everywhere in math, science, and real-world applications. They can model simple relationships between quantities, like the distance traveled at a constant speed or the cost of buying multiple items. Mastering linear equations is a foundational skill for more advanced math topics, so it's worth getting comfortable with them. We can use the order of operations and combining like terms to solve the equation. To solve the equation, we can manipulate it by performing the same operations on both sides. This allows us to isolate the variable and find its value. Linear equations are a fundamental concept in mathematics and have numerous applications in various fields, such as physics, engineering, and economics. They help us model and solve real-world problems involving linear relationships. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The graph of a linear equation is a straight line. Linear equations can have one variable, two variables, or more. The most common form of a linear equation with one variable is ax + b = 0, where a and b are constants and x is the variable. The solution to a linear equation is the value of the variable that makes the equation true. To solve a linear equation, we use algebraic operations to isolate the variable on one side of the equation. This involves adding, subtracting, multiplying, or dividing both sides of the equation by the same value. The goal is to get the variable by itself on one side of the equation, so that we can determine its value.

Step-by-Step Solution

Okay, let's get down to business and solve our equation:

d−10−2d+7=8+d−10−3dd - 10 - 2d + 7 = 8 + d - 10 - 3d

Step 1: Combine Like Terms on Each Side

The first thing we want to do is simplify both sides of the equation by combining like terms. Like terms are those that have the same variable (in this case, 'd') or are constants (just numbers). On the left side, we have 'd' and '-2d', which combine to '-d'. We also have '-10' and '+7', which combine to '-3'. So, the left side becomes:

−d−3-d - 3

On the right side, we have 'd' and '-3d', which combine to '-2d'. We also have '8' and '-10', which combine to '-2'. So, the right side becomes:

−2d−2-2d - 2

Now our equation looks simpler:

−d−3=−2d−2-d - 3 = -2d - 2

Step 2: Move Variable Terms to One Side

Next, we want to get all the terms with 'd' on one side of the equation. It doesn't matter which side, but let's choose the left side for this example. To move the '-2d' term from the right side to the left side, we add '2d' to both sides. Remember, whatever we do to one side, we have to do to the other!

−d−3+2d=−2d−2+2d-d - 3 + 2d = -2d - 2 + 2d

This simplifies to:

d−3=−2d - 3 = -2

Step 3: Move Constant Terms to the Other Side

Now we want to isolate the 'd' term. To do this, we need to move the constant term (-3) to the right side of the equation. We do this by adding '3' to both sides:

d−3+3=−2+3d - 3 + 3 = -2 + 3

This simplifies to:

d=1d = 1

Step 4: Check Your Solution

It's always a good idea to check your solution to make sure it's correct. We do this by plugging our solution (d = 1) back into the original equation and seeing if both sides are equal:

1−10−2(1)+7=8+1−10−3(1)1 - 10 - 2(1) + 7 = 8 + 1 - 10 - 3(1)

Simplifying both sides:

1−10−2+7=8+1−10−31 - 10 - 2 + 7 = 8 + 1 - 10 - 3

−4=−4-4 = -4

Since both sides are equal, our solution is correct!

Combining like terms is a crucial step in solving linear equations. It involves grouping terms with the same variable and constant terms separately. This simplifies the equation and makes it easier to solve. By combining like terms, we reduce the number of terms in the equation, which makes it less complex and more manageable. Adding or subtracting the same value from both sides of the equation maintains the equality. This allows us to isolate the variable on one side of the equation. Multiplying or dividing both sides of the equation by the same non-zero value also preserves the equality. This helps us further isolate the variable and find its value. Checking the solution by substituting it back into the original equation is an important step to ensure accuracy. If the solution does not satisfy the equation, it indicates an error in the solving process, and we need to re-evaluate the steps taken. This helps us identify and correct any mistakes made during the solving process.

The Answer

So, we've solved the equation, and we found that:

d=1d = 1

That means the correct answer is D. d = 1

Give yourself a pat on the back if you followed along and got the same answer! If not, don't worry – the key is practice. The first step in solving the equation is to simplify both sides by combining like terms. This involves grouping the 'd' terms and the constant terms separately. This makes the equation easier to work with. Next, we want to isolate the 'd' term on one side of the equation. To do this, we can add or subtract terms from both sides of the equation. The goal is to get all the 'd' terms on one side and all the constant terms on the other side. Once we have the 'd' term isolated, we can solve for 'd' by dividing both sides of the equation by the coefficient of 'd'. This will give us the value of 'd' that satisfies the equation. It's always a good idea to check our answer by plugging it back into the original equation to make sure it works. This helps us catch any mistakes we might have made along the way.

Tips and Tricks for Solving Linear Equations

Solving linear equations becomes easier with practice, but here are a few extra tips and tricks to keep in mind:

  • Always simplify first: Before you start moving terms around, make sure you've simplified both sides of the equation as much as possible by combining like terms.
  • Do the same thing to both sides: This is the golden rule of equation solving! Any operation you perform on one side of the equation must be performed on the other side to maintain equality.
  • Work backwards: Think about the order of operations (PEMDAS/BODMAS) in reverse. To isolate the variable, you typically undo addition/subtraction first, then multiplication/division.
  • Don't be afraid of fractions: If you end up with fractions, don't panic! You can often clear them by multiplying both sides of the equation by the least common multiple of the denominators.
  • Check your work: We can't stress this enough! Plugging your solution back into the original equation is the best way to catch errors.
  • Practice, practice, practice: The more you solve linear equations, the more comfortable you'll become with the process. Seek out practice problems online or in textbooks.

Solving linear equations involves a systematic approach of isolating the variable by performing operations on both sides of the equation. It's crucial to maintain the balance of the equation by applying the same operations to both sides. This ensures that the solution remains valid. When dealing with more complex equations, it's helpful to break them down into smaller, more manageable steps. This makes the process less overwhelming and reduces the chances of making errors. Linear equations are a fundamental concept in algebra and have numerous real-world applications. Mastering the techniques for solving them is essential for further studies in mathematics and related fields. Remember, practice makes perfect! The more linear equations you solve, the more confident and proficient you'll become. Don't be afraid to seek help from teachers, tutors, or online resources if you encounter difficulties. With consistent effort and the right approach, you can master the art of solving linear equations.

Conclusion

So, there you have it! We've successfully solved the linear equation $d - 10 - 2d + 7 = 8 + d - 10 - 3d$ and found that $d = 1$. Remember, solving linear equations is all about simplifying, isolating the variable, and keeping the equation balanced. With these steps and a little practice, you'll be solving linear equations like a pro in no time. Keep up the great work, guys!