Solving 3x-2y=11 And 5x+3y=-7 By Substitution Method A Step-by-Step Guide
Hey guys! Ever find yourself staring blankly at a system of equations, feeling like you're trying to decipher ancient hieroglyphics? Don't worry, you're not alone! Systems of equations can seem intimidating at first, but with the right approach, they become much less scary. In this guide, we're going to break down the substitution method, a powerful technique for solving these types of problems. We'll use the specific example of 3x - 2y = 11 and 5x + 3y = -7 to illustrate each step clearly. So, grab your pencil and paper, and let's dive in!
Understanding Systems of Equations
Before we jump into the solution, let's make sure we're all on the same page about what a system of equations actually is. Simply put, a system of equations is a set of two or more equations that share the same variables. The goal is to find values for these variables that satisfy all the equations in the system simultaneously. Think of it like finding the perfect combination that unlocks a secret code. In our example, we have two equations, each containing the variables 'x' and 'y'. Our mission is to find the values of 'x' and 'y' that make both equations true at the same time.
Now, there are several methods to solve systems of equations, including graphing, elimination, and, of course, substitution. Each method has its strengths and weaknesses, and the best choice often depends on the specific system you're dealing with. The substitution method is particularly useful when one of the equations can be easily rearranged to isolate one variable in terms of the other. This is exactly the case in our example, which makes substitution a great option. We will discuss the step-by-step process to master substitution using the equations 3x - 2y = 11 and 5x + 3y = -7. Solving systems of equations is a fundamental skill in algebra and has applications in various fields, such as engineering, economics, and computer science. Understanding different methods and knowing when to apply each one is crucial for success in mathematical problem-solving.
Step 1: Choose an Equation and Isolate a Variable
The first step in the substitution method is to pick one of the equations and solve it for one of the variables. The key here is to choose the equation and variable that will be easiest to isolate. In other words, look for a variable that has a coefficient of 1 or -1, as this will minimize the chances of dealing with fractions. Looking at our system:
- 3x - 2y = 11
- 5x + 3y = -7
Neither 'x' nor 'y' has a coefficient of 1 or -1 in either equation. However, we can still make a strategic choice. Let's choose the first equation, 3x - 2y = 11, and solve for 'x'. Why 'x'? Because it has a smaller coefficient (3) than the coefficient of 'y' (-2), which might make the algebra slightly simpler. To isolate 'x', we'll perform the following steps:
- Add 2y to both sides of the equation:
3x - 2y + 2y = 11 + 2y
This simplifies to: 3x = 11 + 2y - Divide both sides by 3:
3x / 3 = (11 + 2y) / 3
This gives us:
x = (11 + 2y) / 3
Now we have successfully isolated 'x' in terms of 'y'. This expression for 'x' is crucial for the next step. Remember, the goal is to express one variable in terms of the other so that we can substitute it into the other equation. By isolating 'x', we've created a pathway to eliminate 'x' from the second equation, leaving us with an equation in just 'y'. This is the essence of the substitution method. Choosing the right variable to isolate can significantly impact the complexity of the problem. While we chose to solve for 'x' in the first equation, you could also choose to solve for 'y' in the first equation or either variable in the second equation. The best choice often comes down to personal preference and recognizing which option will lead to the cleanest algebraic manipulations. Practice is key to developing this intuition. Keep in mind that accuracy is paramount in this step. A mistake in isolating the variable will propagate through the rest of the solution, leading to an incorrect answer. Double-check your work and ensure that you have correctly solved for the chosen variable before moving on.
Step 2: Substitute the Expression into the Other Equation
This is where the magic of substitution really happens! Now that we have an expression for 'x' in terms of 'y', we can substitute it into the other equation. Remember, we used the first equation to solve for 'x', so we must substitute the expression we found into the second equation, which is 5x + 3y = -7. This is crucial; substituting back into the same equation we used to isolate 'x' would just lead us back to where we started and wouldn't help us solve for the variables.
So, we replace 'x' in the second equation with the expression we found in Step 1, which was x = (11 + 2y) / 3. This gives us:
5 * ((11 + 2y) / 3) + 3y = -7
Notice what we've achieved: we've transformed the equation into one that involves only 'y'. This is a significant step forward because we can now solve this single equation for 'y'. This substitution step is the heart of the method, and it's essential to perform it accurately. Double-check that you've correctly replaced the variable with its expression and that you're using the other equation in the system. The next part involves simplifying and solving the resulting equation for 'y'. This will require careful attention to algebraic manipulations, such as distributing, combining like terms, and isolating 'y'. By successfully substituting, we've reduced the system of two equations in two variables to a single equation in one variable, making the problem much more manageable. The skill of substitution is widely applicable in various mathematical contexts, not just in solving systems of equations. It's a fundamental technique that appears in calculus, differential equations, and other advanced topics. Mastering substitution is therefore an investment in your overall mathematical proficiency.
Step 3: Solve for the Remaining Variable
Okay, we've done the hard part of substituting! Now comes the satisfying part where we get to solve for one of the variables. We're currently looking at the equation we obtained after substituting, which is:
5 * ((11 + 2y) / 3) + 3y = -7
Our goal is to isolate 'y' on one side of the equation. To do this, we need to simplify the equation step-by-step. First, let's get rid of the fraction by multiplying every term in the equation by 3:
3 * [5 * ((11 + 2y) / 3) + 3y] = 3 * -7
This simplifies to:
5 * (11 + 2y) + 9y = -21
Now, we distribute the 5:
55 + 10y + 9y = -21
Next, we combine the 'y' terms:
55 + 19y = -21
Now, we subtract 55 from both sides:
19y = -21 - 55
19y = -76
Finally, we divide both sides by 19 to solve for 'y':
y = -76 / 19
y = -4
Woohoo! We've found the value of 'y'. This is a major milestone in solving the system of equations. With the value of 'y' in hand, we're now just one step away from finding the value of 'x'. The process of solving for 'y' demonstrates the importance of careful algebraic manipulation. Each step, from multiplying by 3 to combining like terms and isolating 'y', requires attention to detail and a solid understanding of algebraic principles. A mistake in any of these steps can lead to an incorrect value for 'y', which would then impact the final solution. Double-checking your work at each step is crucial to ensure accuracy. The equation we solved in this step was a linear equation in one variable, which is a fundamental type of equation in algebra. Mastering the techniques for solving these equations is essential for tackling more complex mathematical problems. The value of 'y' we found, -4, is a critical piece of information. It tells us the y-coordinate of the point where the two lines represented by the original equations intersect. This point of intersection is the solution to the system of equations. We're now ready to use this value to find the corresponding x-coordinate.
Step 4: Substitute the Value Back to Find the Other Variable
We're on the home stretch now! We've successfully found that y = -4. The final step is to substitute this value back into either of the original equations (or the equation we derived in Step 1) to solve for 'x'. The choice of which equation to use is up to you; pick the one that looks easier to work with. Let's use the equation we derived in Step 1, where we isolated 'x':
x = (11 + 2y) / 3
Now, we substitute y = -4 into this equation:
x = (11 + 2 * (-4)) / 3
Simplify:
x = (11 - 8) / 3
x = 3 / 3
x = 1
Fantastic! We've found that x = 1. We now have both values, x = 1 and y = -4, which is the solution to our system of equations. This back-substitution step is a crucial part of the substitution method. It allows us to leverage the value we found for one variable to determine the value of the other. The accuracy of this step depends on the correctness of the value we found in the previous step, so it's essential to double-check our work along the way. The equation we chose to substitute into, x = (11 + 2y) / 3, was a convenient choice because 'x' was already isolated. However, we could have also substituted y = -4 into either of the original equations, 3x - 2y = 11 or 5x + 3y = -7, and solved for 'x'. The result would be the same, but the algebraic manipulations might be slightly different. This highlights the flexibility of the substitution method and the importance of choosing a strategy that feels comfortable and efficient for you. The values we found, x = 1 and y = -4, represent the coordinates of the point where the two lines represented by the original equations intersect on a graph. This point (1, -4) is the unique solution that satisfies both equations simultaneously. We've successfully navigated the substitution method and found the solution to our system of equations!
Step 5: Check Your Solution
Always, always, always check your solution! It's the best way to catch any mistakes and ensure that you've got the correct answer. To check our solution, we substitute the values we found, x = 1 and y = -4, back into the original equations:
- Equation 1: 3x - 2y = 11 Substitute: 3 * (1) - 2 * (-4) = 11 Simplify: 3 + 8 = 11 11 = 11 (This is true!)
- Equation 2: 5x + 3y = -7 Substitute: 5 * (1) + 3 * (-4) = -7 Simplify: 5 - 12 = -7 -7 = -7 (This is also true!)
Since our values for 'x' and 'y' satisfy both original equations, we can confidently say that our solution is correct. The solution to the system of equations is x = 1 and y = -4, or as an ordered pair, (1, -4).
Checking your solution is a vital step in the problem-solving process, not just in mathematics but in many areas of life. It's a way to verify your work, identify any errors, and gain confidence in your answer. In the context of systems of equations, checking ensures that the values you found satisfy all the equations in the system simultaneously. This is the ultimate test of whether your solution is correct. The process of checking involves substituting the values back into the original equations, which allows you to retrace your steps and identify any potential mistakes in your algebraic manipulations. If the values do not satisfy one or both equations, it means there's an error somewhere in your solution process, and you need to go back and review your work. The fact that our values satisfied both equations confirms that we correctly applied the substitution method and arrived at the accurate solution. The ordered pair (1, -4) represents the point of intersection of the two lines represented by the original equations. This point is the unique solution that lies on both lines, satisfying both equations. By checking our solution, we've not only verified our answer but also deepened our understanding of what it means to solve a system of equations. We can now confidently move on to other problems, knowing that we have a reliable method for finding solutions and a way to check our work.
Conclusion
And there you have it! We've successfully solved the system of equations 3x - 2y = 11 and 5x + 3y = -7 using the substitution method. We found that x = 1 and y = -4, and we even checked our solution to make sure it was correct. Remember, the key to mastering the substitution method is to break it down into manageable steps: isolate a variable, substitute the expression, solve for the remaining variable, substitute back to find the other variable, and always check your solution. With practice, you'll become a pro at solving systems of equations! Keep practicing, and don't be afraid to tackle those tricky problems. You've got this!
