Solve Linear Equations: A Step-by-Step Guide

Jan 8, 2026 by Andrew McMorgan 45 views

Hey guys! Today, we're diving deep into the world of algebra, specifically tackling how to solve linear equations. You know, those problems that look a bit intimidating at first but are totally manageable when you break them down. We'll walk through a complete example, solving the equation $3x - 6(5x + 3) = 9x + 6$ step-by-step. So grab your notebooks, get comfy, and let's get this math party started!

Understanding Linear Equations

First off, what exactly is a linear equation? In simple terms, it's an equation where the highest power of the variable (usually 'x') is one. Think of it as a straight line if you were to graph it. Solving these equations means finding the specific value of the variable that makes the equation true. It's like finding the secret key to unlock the balance of the equation. The equation we're working with today, $3x - 6(5x + 3) = 9x + 6$ , is a perfect example of a linear equation that requires a few different algebraic properties to solve. We'll be using the distributive property, combining like terms, and the addition and subtraction properties of equality. These properties are your best friends in algebra because they allow you to manipulate the equation without changing the fundamental truth it represents. Remember, whatever you do to one side of the equation, you must do to the other side to keep it balanced. It’s all about maintaining that equilibrium!

Step 1: Applying the Distributive Property

Alright, let's kick things off with our first step in solving $3x - 6(5x + 3) = 9x + 6$ . The very first thing we need to tackle is that pesky bit in the parentheses: $-6(5x + 3)$ . This is where the distributive property comes into play. This property tells us that when you multiply a number by a sum or difference inside parentheses, you multiply that number by each term inside the parentheses. So, we'll distribute the -6 to both the 5x and the 3.

Calculation:

Multiply -6 by 5x: $-6 \times 5x = -30x$
Multiply -6 by 3: $-6 \times 3 = -18$

Now, we substitute these results back into the original equation:

$3x + (-30x) + (-18) = 9x + 6$

Which simplifies to:

$3x - 30x - 18 = 9x + 6$

See? That scary-looking parenthesis is now gone, replaced by two simpler terms. This step is crucial because it gets rid of the parentheses and exposes more terms that we can work with. Always look for opportunities to use the distributive property when you see a number multiplied by an expression in parentheses. It's a fundamental tool for simplifying equations and making them easier to handle. Don't be afraid of the negative signs; just treat them as part of the number you're distributing. It's like carefully unwrapping a present – you have to handle all the parts correctly to get to the core.

Step 2: Combining Like Terms

Moving on to step two, guys! Now that we've applied the distributive property, our equation looks like this: $3x - 30x - 18 = 9x + 6$ . Our next mission is to combine like terms. What does that mean? It means grouping together terms that have the same variable raised to the same power. In this equation, the terms with 'x' are $3x$ and $-30x$ . These are our 'like terms' on the left side of the equation.

Calculation:

Combine the 'x' terms on the left side: $3x - 30x = -27x$

The terms $-18$ on the left and $9x$ and $6$ on the right don't have any other like terms to combine with on their respective sides just yet. So, after combining the like terms on the left, our equation becomes:

$-27x - 18 = 9x + 6$

This step is super important because it reduces the number of terms in the equation, making it much cleaner and simpler to solve. The goal is always to simplify, simplify, simplify! By combining $3x$ and $-30x$ , we've taken two terms and made them one, drastically simplifying the left side. It's like tidying up your room – the more you organize, the easier it is to find what you're looking for. Remember, you can only combine terms that are truly alike. You can't mix apples and oranges, and you can't mix 'x' terms with constant terms. Keep those categories separate!

Step 3: Using the Addition Property of Equality

Okay, we're on step three, and our equation is now $-27x - 18 = 9x + 6$ . The goal now is to get all the 'x' terms on one side of the equation and all the constant terms (the numbers without variables) on the other. To do this, we'll use the addition property of equality. This property states that if you add the same value to both sides of an equation, the equation remains balanced. We have an 'x' term on both sides ( $-27x$ and $9x$ ). It's generally easier to move the 'x' term with the smaller coefficient, but either way works. Let's choose to move the $9x$ from the right side to the left side.

Calculation:

To eliminate $9x$ from the right side, we subtract $9x$ from both sides:

$-27x - 18 - 9x = 9x + 6 - 9x$

Now, combine like terms on both sides:

On the left: $-27x - 9x = -36x$

On the right: $9x - 9x = 0$ , leaving just $6$ .

So, the equation becomes:

$-36x - 18 = 6$

Correction: It looks like there was a slight misstep in the original prompt's step 3 calculation. Let's re-evaluate applying the Addition Property of Equality properly to reach the state that leads to the subtraction step. A more common strategy is to move the variable terms to one side and the constants to the other. Let's try moving the $-27x$ to the right by adding $27x$ to both sides, as this will result in a positive coefficient for x, which some find easier to work with.

Starting from: $-27x - 18 = 9x + 6$

Revised Calculation (Adding 27x to both sides):

$-27x - 18 + 27x = 9x + 6 + 27x$

$-18 = (9x + 27x) + 6$

$-18 = 36x + 6$

This revised step aligns with the prompt's stated outcome for step 3, which is $-18 = 36x + 6$ . This is a great example of how different approaches can lead to the same intermediate form. The key is consistency and correct application of the properties. We've successfully isolated the term with 'x' on one side and the constants on the other, even if there's still a constant term on the 'x' side. This is a huge leap towards finding our final answer!

Step 4: Applying the Subtraction Property of Equality

We're now at step four, with our equation standing at $-18 = 36x + 6$ . Our goal is to isolate the term containing 'x' ( $36x$ ) completely. Right now, it has a $+6$ with it. To get rid of that $+6$ , we'll use the subtraction property of equality. Just like the addition property, this rule says that if you subtract the same value from both sides of an equation, it stays balanced. We want to subtract 6 from both sides to cancel out the +6 on the right.

Calculation:

Subtract 6 from both sides of the equation:

$-18 - 6 = 36x + 6 - 6$

Perform the subtraction on both sides:

On the left: $-18 - 6 = -24$

On the right: $6 - 6 = 0$ , leaving just $36x$ .

So, the equation simplifies to:

$-24 = 36x$

Boom! We've managed to get the 'x' term all by itself on one side. This is fantastic progress. This step cleared away the constant term from the side where our variable resided, setting us up perfectly for the final step. It’s like clearing the last obstacle on a race track; you can see the finish line now!

Step 5: The Division Property of Equality (Finding the Solution)

We've reached the final frontier, guys! Our equation is currently $-24 = 36x$ . We're so close to finding the value of 'x'. The variable 'x' is being multiplied by 36. To isolate 'x' completely, we need to undo that multiplication. We do this using the division property of equality. This property states that if you divide both sides of an equation by the same non-zero number, the equation remains balanced. Since 'x' is multiplied by 36, we will divide both sides by 36.

Calculation:

Divide both sides of the equation by 36:

$rac{-24}{36} = rac{36x}{36}$

Simplify both sides:

On the left: $\frac{-24}{36}$ . This fraction can be simplified. Both 24 and 36 are divisible by 12. So, $-24 \div 12 = -2$ and $36 \div 12 = 3$ . The simplified fraction is $-\frac{2}{3}$ .

On the right: $\frac{36x}{36} = x$

So, the solution is:

$x = -\frac{2}{3}$

Verification (Optional but Recommended!):

To be absolutely sure, we can plug this value of x back into the original equation $3x - 6(5x + 3) = 9x + 6$ to see if it holds true.

Left side: $3(-\frac{2}{3}) - 6(5(-\frac{2}{3}) + 3)$

$= -2 - 6(-\frac{10}{3} + \frac{9}{3})$

$= -2 - 6(-\frac{1}{3})$

$= -2 + 2 = 0$

Right side: $9(-\frac{2}{3}) + 6$

$= -6 + 6 = 0$

Since both sides equal 0, our solution $x = -\frac{2}{3}$ is correct! You've successfully navigated the entire process, from distribution to finding the final value of x. High five!

Conclusion: Mastering Algebraic Equations

And there you have it, folks! We've successfully solved the linear equation $3x - 6(5x + 3) = 9x + 6$ by following a clear, step-by-step process. We used the distributive property to simplify the expression, combined like terms to reduce the number of terms, and then applied the addition property of equality and the subtraction property of equality to gather variable terms on one side and constants on the other. Finally, we used the division property of equality to isolate 'x' and find our solution, $x = -\frac{2}{3}$ .

Solving equations like these is a fundamental skill in mathematics, and the more you practice, the more intuitive it becomes. Remember the core principles: keep the equation balanced by performing the same operation on both sides, and always aim to simplify. Don't get discouraged if you make a mistake; just go back, check your steps, and try again. Math is a journey of problem-solving, and every equation you solve builds your confidence and skill. Keep practicing, keep asking questions, and you'll become an algebra whiz in no time. Until next time, happy calculating!