Solving 2x + 3y = 7 And X - 2y = -7 Step-by-Step Guide
Hey guys! Today, we're diving deep into the world of solving systems of equations. If you've ever felt a bit lost when faced with two equations and two unknowns, don't worry – you're definitely not alone! We're going to break down a classic example step-by-step: solving the system of equations 2x + 3y = 7 and x - 2y = -7. This isn't just about getting the right answer; it's about understanding the why behind the how. So, grab your pencils, and let's get started!
Understanding Systems of Equations
Before we jump into the nitty-gritty of solving 2x + 3y = 7 and x - 2y = -7, let's take a moment to understand what a system of equations actually is. Think of it as a puzzle where you have multiple pieces of information (equations) that you need to fit together to find a solution. In this case, we have two linear equations, each representing a straight line on a graph. The solution to the system is the point where these lines intersect. This point gives us the values of 'x' and 'y' that satisfy both equations simultaneously.
Why are systems of equations important? Well, they pop up all over the place in real-world problems! From calculating the break-even point for a business to figuring out the optimal mix of ingredients in a recipe, systems of equations are a powerful tool for modeling and solving problems. They're not just some abstract math concept; they're a practical skill that can help you make better decisions in various situations.
There are several methods for solving systems of equations, and we'll be focusing on two popular ones in this guide: substitution and elimination. Each method has its own strengths and weaknesses, and understanding both will give you a more flexible approach to problem-solving. The goal is always the same: to find the values of the variables that make all the equations in the system true. So, as we work through solving 2x + 3y = 7 and x - 2y = -7, keep in mind that we're essentially looking for the 'x' and 'y' values that make both of these equations happy!
Method 1: The Substitution Method
The substitution method is a great way to tackle systems of equations, especially when one of the equations is easily solved for one variable in terms of the other. The basic idea is to isolate one variable in one equation and then substitute that expression into the other equation. This eliminates one variable, leaving us with a single equation that we can solve for the remaining variable. Let's see how this works in practice with our example, solving 2x + 3y = 7 and x - 2y = -7.
Our first step is to choose one of the equations and solve it for one of the variables. Looking at our equations, x - 2y = -7 seems like a good candidate because it's relatively easy to isolate 'x'. By adding 2y to both sides, we get x = 2y - 7. Now we have an expression for 'x' in terms of 'y'. This is the key to the substitution method – we've found a way to represent 'x' using 'y'.
Next, we take this expression for 'x' and substitute it into the other equation. We have 2x + 3y = 7, and we know that x = 2y - 7, so we can substitute (2y - 7) in place of 'x'. This gives us 2(2y - 7) + 3y = 7. Notice that we've now eliminated 'x' and have an equation with only 'y' as the variable. This is exactly what we wanted!
Now we can solve this equation for 'y'. First, distribute the 2: 4y - 14 + 3y = 7. Then, combine like terms: 7y - 14 = 7. Add 14 to both sides: 7y = 21. Finally, divide by 7: y = 3. We've found the value of 'y'! But we're not done yet – we still need to find 'x'.
To find 'x', we can substitute the value of 'y' (which is 3) back into either of our original equations or the expression we found for 'x' earlier. Using x = 2y - 7 is the easiest route. Substituting y = 3 gives us x = 2(3) - 7, which simplifies to x = 6 - 7, so x = -1. We've now found both 'x' and 'y'!
Therefore, the solution to the system of equations 2x + 3y = 7 and x - 2y = -7 is x = -1 and y = 3. This means that the point (-1, 3) is the intersection point of the two lines represented by these equations. We've successfully used the substitution method to solve the system! Remember, the key to this method is to isolate one variable and substitute its expression into the other equation. With practice, you'll become a substitution master!
Method 2: The Elimination Method
The elimination method, also known as the addition method, is another powerful technique for solving systems of equations. This method is particularly useful when the coefficients of one of the variables in the two equations are either the same or easily made the same (or opposites) by multiplying one or both equations by a constant. The core idea behind elimination is to manipulate the equations so that when you add them together, one of the variables cancels out, leaving you with a single equation in one variable. Let's apply this method to our example, solving 2x + 3y = 7 and x - 2y = -7.
Our goal is to eliminate either 'x' or 'y'. Looking at the equations, it seems easier to eliminate 'x' because we can multiply the second equation by -2 to get the 'x' terms to be opposites. So, let's multiply the entire equation x - 2y = -7 by -2. This gives us -2x + 4y = 14. Now we have two equations:
- 2x + 3y = 7
- -2x + 4y = 14
Notice that the coefficients of 'x' are now 2 and -2, which are opposites. This is exactly what we wanted! Now we can add the two equations together. When we add the left-hand sides, the 2x and -2x terms cancel out, which eliminates 'x'. Adding the rest of the terms, we get (3y + 4y) = (7 + 14), which simplifies to 7y = 21. We've successfully eliminated 'x' and are left with an equation in just 'y'.
Now we can solve this equation for 'y'. Dividing both sides by 7, we get y = 3. We've found the value of 'y'! Just like in the substitution method, we're not done yet – we still need to find 'x'.
To find 'x', we can substitute the value of 'y' (which is 3) back into either of our original equations. Let's use the first equation, 2x + 3y = 7. Substituting y = 3 gives us 2x + 3(3) = 7, which simplifies to 2x + 9 = 7. Subtracting 9 from both sides, we get 2x = -2. Finally, dividing by 2, we get x = -1. We've found both 'x' and 'y'!
Therefore, using the elimination method, we've also found that the solution to the system of equations 2x + 3y = 7 and x - 2y = -7 is x = -1 and y = 3. This matches the solution we found using the substitution method, which confirms that our solution is correct. The elimination method is a powerful tool when you can easily manipulate the equations to eliminate a variable. Remember, the key is to find a way to make the coefficients of one of the variables opposites so that they cancel out when you add the equations together.
Verifying the Solution
Okay, guys, we've gone through the process of solving 2x + 3y = 7 and x - 2y = -7 using both the substitution and elimination methods, and we arrived at the solution x = -1 and y = 3. But how can we be absolutely sure that our solution is correct? That's where verification comes in! Verifying our solution is a crucial step in solving systems of equations because it helps us catch any potential errors we might have made along the way. It's like double-checking your work to make sure everything adds up.
The process of verifying a solution is quite simple. All we need to do is substitute the values we found for x and y back into the original equations and see if they hold true. If both equations are satisfied by our solution, then we know we've found the correct answer. If even one equation is not satisfied, it means we made a mistake somewhere in our calculations, and we need to go back and check our work.
Let's start by substituting x = -1 and y = 3 into the first equation, 2x + 3y = 7. Substituting these values, we get 2(-1) + 3(3) = 7. Simplifying, we have -2 + 9 = 7, which is indeed true. So, our solution satisfies the first equation.
Now, let's do the same for the second equation, x - 2y = -7. Substituting x = -1 and y = 3, we get (-1) - 2(3) = -7. Simplifying, we have -1 - 6 = -7, which is also true. So, our solution satisfies the second equation as well.
Since our solution x = -1 and y = 3 satisfies both equations in the system, we can confidently say that it is the correct solution. Verifying the solution is a simple but essential step that ensures the accuracy of our answer. It's always a good practice to verify your solutions, especially in more complex problems, to avoid careless errors. So, next time you're solving systems of equations, don't forget to verify your solution – it's the ultimate way to know you've got it right!
Choosing the Best Method
We've now explored two powerful methods for solving systems of equations: substitution and elimination. We even used them both to tackle the same problem, solving 2x + 3y = 7 and x - 2y = -7, and we arrived at the same solution each time (which is always a good sign!). But you might be wondering,
