Solving Linear Equations: A Step-by-Step Guide

Hey guys! Let's dive into the world of solving linear equations. We're going to break down how to tackle a system of two equations with two variables. In this case, our equations are: -13y + 11x = -163 and -8x + 7y = 94. Don't worry if this looks a little intimidating at first; we'll go through it step by step, making it easy to understand. We'll explore different methods like substitution or elimination to find the values of x and y that satisfy both equations simultaneously. This is a fundamental concept in algebra, and mastering it will set you up for success in more advanced math topics. We'll start with the basics, ensuring you understand each step. So, grab your pencils and let's get started! Understanding how to solve these equations is crucial for a wide range of applications, from everyday problem-solving to complex scientific calculations. We'll also talk about how to verify our solutions, ensuring accuracy. By the end of this guide, you'll be able to confidently solve similar problems. This method is used in various fields, including computer science, engineering, and economics. Let's make learning math a little less stressful and a lot more fun, shall we? You'll find that with practice, these types of problems become much easier. Remember, practice makes perfect! So, let's jump right into the first method. It will also help you to think critically and analytically, skills that are invaluable in all areas of life. This guide is designed to be accessible to everyone, regardless of their prior math experience. We'll make sure to cover all the bases.

Method 1: Elimination

Alright, let's kick things off with the elimination method. The core idea here is to manipulate our equations so that when we add or subtract them, one of the variables vanishes. This simplifies the problem, allowing us to solve for the remaining variable. In our case, we have -13y + 11x = -163 and -8x + 7y = 94. The goal is to make the coefficients of either x or y opposites. Let's aim to eliminate x. To do this, we need to find a common multiple for the coefficients of x, which are 11 and -8. The least common multiple (LCM) of 11 and 8 is 88. So, we'll multiply the first equation by 8 and the second equation by 11. Here's what that looks like: First Equation: 8 * (-13y + 11x) = 8 * (-163) which simplifies to -104y + 88x = -1304. Second Equation: 11 * (-8x + 7y) = 11 * (94) which simplifies to -88x + 77y = 1034. Now, notice the coefficients of x are +88 and -88. When we add these two equations together, the x terms will cancel out! Let's add them: (-104y + 88x) + (-88x + 77y) = -1304 + 1034. This simplifies to -27y = -270. Now, to find y, we divide both sides by -27: y = -270 / -27 which gives us y = 10. Yay! We've found the value of y. Now, to find x, we can substitute this value back into either of the original equations. Let's use the second equation: -8x + 7y = 94. Substitute y = 10: -8x + 7(10) = 94. Simplify: -8x + 70 = 94. Subtract 70 from both sides: -8x = 24. Divide both sides by -8: x = 24 / -8, which gives us x = -3. So, we have found that x = -3 and y = 10. But hold on, we're not done yet! We need to verify that these values satisfy both original equations. Let's plug them into the original equations to check if our answers are correct. Always verify your answers to avoid errors!

Verifying the Solution

Okay, let's verify our solution (x = -3, y = 10). We'll plug these values into our original equations to ensure they hold true. Let's start with the first equation: -13y + 11x = -163. Substituting x = -3 and y = 10, we get: -13(10) + 11(-3) = -130 - 33 = -163. And it checks out! Now, let's verify the second equation: -8x + 7y = 94. Substituting x = -3 and y = 10, we get: -8(-3) + 7(10) = 24 + 70 = 94. And it also checks out! Both equations are true with these values, which confirms that our solution (x = -3, y = 10) is correct. Awesome job, guys! This process is crucial to ensure that we've found the right values for our variables. Remember, making a small mistake somewhere can lead to an incorrect answer, so verification is an essential part of solving these types of problems. This also helps to build your confidence and helps you avoid silly mistakes. Always double-check your solutions. This step might seem like a bit of extra work, but it can save you a lot of time and trouble in the long run. By consistently verifying your answers, you'll become more confident in your ability to solve these types of problems. Also, you learn how to identify your mistake. And now, let's explore another method.

Method 2: Substitution

Alright, let's switch gears and explore the substitution method. This method involves solving one of the equations for one variable in terms of the other, and then substituting that expression into the other equation. Let's use our same equations: -13y + 11x = -163 and -8x + 7y = 94. First, let's solve the second equation for x. Rearranging -8x + 7y = 94, we get: -8x = -7y + 94. Divide both sides by -8: x = (7/8)y - (94/8). Simplify: x = (7/8)y - (47/4). Now, substitute this expression for x into the first equation: -13y + 11[(7/8)y - (47/4)] = -163. Distribute the 11: -13y + (77/8)y - (517/4) = -163. To combine the y terms, we need a common denominator, which is 8. So, convert -13y to -104/8y. Our equation is now: (-104/8)y + (77/8)y - (517/4) = -163. Combine the y terms: (-27/8)y - (517/4) = -163. Add (517/4) to both sides: (-27/8)y = -163 + (517/4). Convert -163 to a fraction with a denominator of 4: -163 = -652/4. So, we now have: (-27/8)y = (-652/4) + (517/4). Simplify: (-27/8)y = -135/4. Multiply both sides by -8/27: y = (-135/4) * (-8/27). Simplify: y = 10. Great, we have y = 10 again! Now, substitute y = 10 back into the expression we found for x: x = (7/8)y - (47/4). So, x = (7/8)(10) - (47/4). Simplify: x = 70/8 - 47/4. Convert 47/4 to have a denominator of 8: 47/4 = 94/8. So, x = 70/8 - 94/8. x = -24/8 = -3. So, we get x = -3, and y = 10, the same solution we found using elimination! We have a second method to verify our solutions. Remember that different methods can be used to solve the same problem.

Comparing the Methods

So, guys, you've now seen two methods: elimination and substitution. Which one is