Solving Linear Equations A Step-by-Step Guide To -7x + 12 - 2x = 23 + 13x

Solving linear equations is a fundamental skill in algebra, serving as a cornerstone for more advanced mathematical concepts. This article provides a detailed, step-by-step guide on how to solve the equation $-7x + 12 - 2x = 23 + 13x$. We will break down each step, ensuring a clear understanding of the underlying principles. This comprehensive approach aims to equip you with the knowledge and confidence to tackle similar problems effectively. Our focus will be on clarity, accuracy, and a human-centered explanation that makes the process intuitive and straightforward.

Step 1: Simplifying by Combining Like Terms

The initial step in solving the equation involves simplifying both sides by combining like terms. Like terms are those that contain the same variable raised to the same power (e.g., $-7x$ and $-2x$) or are constants (e.g., $12$, $23$). The importance of this step cannot be overstated, as it reduces the complexity of the equation, making it easier to manipulate and solve. By grouping like terms, we consolidate the equation into a more manageable form. This process not only simplifies the equation but also sets the stage for subsequent steps, such as isolating the variable. Combining like terms is a core algebraic technique, crucial for solving various types of equations, including linear, quadratic, and polynomial equations. The meticulous execution of this step minimizes the chances of errors and lays a solid foundation for the rest of the solution.

Identifying Like Terms on the Left Side

On the left side of the equation, we have $-7x + 12 - 2x$. The like terms here are $-7x$ and $-2x$. These terms both contain the variable $x$ raised to the power of 1, making them eligible for combination. The constant term, $12$, stands alone on this side and will be dealt with in a later step. It's crucial to correctly identify these like terms because any mistake at this stage can propagate through the rest of the solution, leading to an incorrect answer. Think of this step as organizing your tools before starting a project; it's about putting everything in its right place to make the job smoother. In the context of algebra, this means grouping similar elements together to simplify the equation.

Combining $-7x$ and $-2x$

To combine $-7x$ and $-2x$, we simply add their coefficients. The coefficient of a term is the numerical part of the term that multiplies the variable. In this case, the coefficients are $-7$ and $-2$. Adding these together, we get $-7 + (-2) = -9$. Therefore, the combined term is $-9x$. This arithmetic operation is a fundamental part of algebra and is used extensively in simplifying expressions and solving equations. Mastering the addition and subtraction of negative numbers is essential for accurately combining like terms. The result, $-9x$, represents a single term that replaces the two original terms, making the equation more concise.

Rewriting the Left Side

After combining $-7x$ and $-2x$ into $-9x$, the left side of the equation becomes $-9x + 12$. This simplified form is much easier to work with. By reducing the number of terms, we make the equation less cluttered and easier to understand. The constant term, $12$, remains unchanged as there are no other constant terms on the left side to combine it with. This step is akin to tidying up a workspace; by removing clutter, we create a clearer path to the solution. The simplified left side, $-9x + 12$, is now ready for further manipulation in the subsequent steps of solving the equation. It is crucial to double-check this step to ensure that the like terms have been combined correctly.

Identifying Like Terms on the Right Side

Now, let's consider the right side of the equation, which is $23 + 13x$. On this side, we have a constant term, $23$, and a term with the variable $x$, which is $13x$. In this case, there are no like terms to combine on the right side. The term $13x$ is the only term with the variable $x$, and $23$ is the only constant term. Therefore, the right side of the equation remains as $23 + 13x$. Recognizing when terms cannot be combined is just as important as knowing when they can be. This understanding helps prevent unnecessary steps and keeps the solution process efficient.

Maintaining the Equation's Balance

It's important to remember that whatever operation we perform on one side of the equation, we must perform the same operation on the other side to maintain the equation's balance. This principle is fundamental to solving equations. The equal sign (=) signifies that the expressions on both sides have the same value. Any operation that disrupts this balance will lead to an incorrect solution. Therefore, after simplifying one side, we ensure that the other side is also addressed appropriately. In this case, since the right side did not require simplification, we simply acknowledge it and move on to the next step. The balance of the equation is a guiding principle throughout the entire solution process.

Updated Equation

After simplifying the left side, our equation now looks like this: $-9x + 12 = 23 + 13x$. This is a more streamlined version of the original equation, making it easier to proceed with the next steps. By combining like terms, we have reduced the complexity and set the stage for isolating the variable $x$. The updated equation serves as a clear checkpoint, allowing us to review the progress made and ensure that the simplification was performed correctly. From here, we can move forward with confidence, knowing that we have a solid foundation to build upon.

Step 2: Isolating the Variable Term

Isolating the variable term is a crucial step in solving equations. It involves moving all terms containing the variable to one side of the equation and all constant terms to the other side. This process helps to eventually get the variable by itself, which is the ultimate goal. Think of it as sorting puzzle pieces; you need to group similar pieces together before you can see the whole picture. In algebraic terms, grouping variable terms and constant terms allows us to simplify the equation and reveal the value of the variable. The technique used here is based on the properties of equality, which allow us to perform the same operation on both sides of the equation without changing its solution.

Choosing a Side for the Variable

To begin isolating the variable term, we need to decide which side of the equation to move the variable terms to. In our equation, $-9x + 12 = 23 + 13x$, we have variable terms on both sides: $-9x$ on the left and $13x$ on the right. There's no strict rule on which side to choose, but it's often easier to move the term with the smaller coefficient to the side with the larger coefficient. This can help avoid dealing with negative coefficients, which some people find more confusing. In this case, $-9$ is smaller than $13$, so it might be advantageous to move the $-9x$ term to the right side. However, for the sake of demonstration, let’s choose to move the $13x$ term to the left side.

Subtracting $13x$ from Both Sides

To move $13x$ from the right side to the left side, we perform the inverse operation, which is subtraction. We subtract $13x$ from both sides of the equation. This ensures that we maintain the equation's balance. Subtracting $13x$ from the right side cancels out the $13x$ term, leaving us with just the constant term $23$. On the left side, we subtract $13x$ from $-9x$. The operation looks like this: $-9x - 13x$. To perform this subtraction, we combine the coefficients: $-9 - 13 = -22$. Therefore, $-9x - 13x$ simplifies to $-22x$. This step is a direct application of the subtraction property of equality, which states that subtracting the same value from both sides of an equation preserves the equality.

Updated Equation with Variable Terms on One Side

After subtracting $13x$ from both sides, our equation now looks like this: $-22x + 12 = 23$. Notice that all the terms with the variable $x$ are now on the left side, which is what we intended. The right side now contains only the constant term $23$. This is a significant step forward in isolating the variable. The equation is now in a form where we can focus on isolating the variable term, $-22x$. This consolidation of variable terms simplifies the subsequent steps and brings us closer to the final solution. It is crucial to review this step to ensure that the subtraction was performed correctly and that the equation remains balanced.

Moving the Constant Term

Now that the variable terms are on one side, the next step is to move the constant term from the variable side to the other side. In our equation, $-22x + 12 = 23$, the constant term on the variable side is $12$. To move it to the right side, we perform the inverse operation, which is subtraction. We subtract $12$ from both sides of the equation. This maintains the equation's balance and isolates the variable term further.

Subtracting 12 from Both Sides

Subtracting $12$ from both sides of the equation, we get: $-22x + 12 - 12 = 23 - 12$. On the left side, the $+12$ and $-12$ cancel each other out, leaving us with just the term $-22x$. On the right side, we perform the subtraction: $23 - 12 = 11$. This step is another application of the subtraction property of equality. By subtracting the same value from both sides, we ensure that the equality is preserved. The arithmetic operation here is straightforward, but it's essential to perform it accurately to avoid errors.

Equation with Isolated Variable Term

After subtracting $12$ from both sides, our equation now looks like this: $-22x = 11$. This is a critical milestone in solving the equation. We have successfully isolated the variable term, $-22x$, on the left side, and we have a constant term, $11$, on the right side. The equation is now in a very simple form, making it easy to solve for $x$. This step highlights the power of inverse operations in isolating variables. By strategically adding or subtracting terms, we can manipulate the equation to reveal the solution. The equation $-22x = 11$ is now ready for the final step: solving for $x$.

Step 3: Solving for $x$

The final step in solving the equation is to isolate $x$ completely. We have the equation $-22x = 11$. Here, $x$ is being multiplied by $-22$. To isolate $x$, we need to perform the inverse operation of multiplication, which is division. This step involves dividing both sides of the equation by the coefficient of $x$. The coefficient is the number that multiplies the variable, in this case, $-22$. Dividing both sides by the coefficient will give us the value of $x$. This step is a direct application of the division property of equality, which is a fundamental principle in solving equations.

Dividing Both Sides by -22

To isolate $x$, we divide both sides of the equation $-22x = 11$ by $-22$. This gives us: rac{-22x}{-22} = rac{11}{-22}. On the left side, the $-22$ in the numerator and the $-22$ in the denominator cancel each other out, leaving us with just $x$. On the right side, we perform the division: rac{11}{-22}. This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is $11$. Dividing $11$ by $11$ gives us $1$, and dividing $-22$ by $11$ gives us $-2$. Therefore, the simplified fraction is rac{1}{-2}, which is equal to -rac{1}{2}.

The Solution

After dividing both sides by $-22$, we find that x = -rac{1}{2}. This is the solution to the equation $-7x + 12 - 2x = 23 + 13x$. The solution means that if we substitute -rac{1}{2} for $x$ in the original equation, both sides of the equation will be equal. This is the final answer, and it represents the value of $x$ that satisfies the equation. We have successfully solved for $x$ by systematically applying algebraic principles and inverse operations. The solution x = -rac{1}{2} can be further verified by substituting it back into the original equation.

Verification of the Solution

To verify our solution, we substitute x = -rac{1}{2} back into the original equation, $-7x + 12 - 2x = 23 + 13x$. This involves replacing every instance of $x$ in the equation with -rac{1}{2}. This step is crucial to ensure that our solution is correct. If both sides of the equation are equal after the substitution, then our solution is verified. If they are not equal, it indicates that an error was made somewhere in the solution process, and we need to go back and check our work. Verification is a safeguard against mistakes and ensures the accuracy of our result.

Substituting x = -rac{1}{2} into the Original Equation

Substituting x = -rac{1}{2} into the original equation, we get: -7(-rac{1}{2}) + 12 - 2(-rac{1}{2}) = 23 + 13(-rac{1}{2}). Now, we simplify both sides of the equation. On the left side, -7(-rac{1}{2}) is equal to rac{7}{2}, and -2(-rac{1}{2}) is equal to $1$. So, the left side becomes: rac{7}{2} + 12 + 1. On the right side, 13(-rac{1}{2}) is equal to -rac{13}{2}. So, the right side becomes: 23 - rac{13}{2}. Next, we need to express all terms with a common denominator to perform the addition and subtraction. The common denominator here is $2$.

Simplifying Both Sides

Converting all terms to a common denominator of $2$, the left side becomes: rac{7}{2} + rac{24}{2} + rac{2}{2}. Adding these fractions, we get: rac{7 + 24 + 2}{2} = rac{33}{2}. On the right side, we convert $23$ to a fraction with a denominator of $2$: rac{46}{2}. So, the right side becomes: rac{46}{2} - rac{13}{2}. Subtracting these fractions, we get: rac{46 - 13}{2} = rac{33}{2}.

Conclusion of Verification

After simplifying both sides of the equation, we find that the left side, rac{33}{2}, is equal to the right side, rac{33}{2}. This confirms that our solution, x = -rac{1}{2}, is correct. The verification process demonstrates the importance of checking our work to ensure accuracy. By substituting the solution back into the original equation and verifying that both sides are equal, we can be confident in our answer. This step-by-step approach to solving and verifying equations builds a solid foundation for more advanced algebraic concepts.

Conclusion

In summary, solving the equation $-7x + 12 - 2x = 23 + 13x$ involves several key steps: combining like terms, isolating the variable term, solving for $x$, and verifying the solution. By systematically applying these steps, we can solve linear equations accurately and efficiently. This article has provided a detailed guide to each step, emphasizing the importance of understanding the underlying principles and performing each operation carefully. Mastering these techniques is essential for success in algebra and beyond. Remember, practice is key. The more you solve equations, the more comfortable and confident you will become. Linear equations are a fundamental building block in mathematics, and the ability to solve them is a valuable skill that will serve you well in various fields.