Solving Systems Of Equations A Step-by-Step Guide

Hey guys! Today, we're going to dive into the fascinating world of solving systems of equations. Specifically, we'll tackle a problem that looks like this: $\left\{\begin{aligned}-5 x+y & =3 \\ 10 x-3 y & =-5\end{aligned}\right.$. Don't worry if it seems intimidating at first glance; we'll break it down into manageable steps and you'll be solving these like a pro in no time. Our goal is to find the values of $x$ and $y$ that satisfy both equations simultaneously. We will explore this using the substitution method and the elimination method, providing you with a comprehensive understanding.

Understanding Systems of Equations

Before we jump into solving, let's understand what a system of equations really is. Think of it as a puzzle where you have multiple pieces (equations) and you need to find the solution (values for the variables) that fits all the pieces together. In our 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, represented as $(x, y)$, satisfies both equations. Now, why is this important? Well, systems of equations pop up in various real-world scenarios, from calculating mixtures in chemistry to determining optimal pricing strategies in business. They're a fundamental tool in mathematics and its applications. To get a solid grasp, remember that each equation provides a constraint, and we're looking for the values that meet all the constraints at once. We aim to find the unique pair of $(x, y)$ values that make both equations true. Linear equations, in particular, are equations where the highest power of any variable is 1. This means we won't see terms like $x^2$ or $y^3$. The graphs of linear equations are straight lines, making them relatively straightforward to visualize and solve. Mastering the techniques to solve these systems is crucial for advancing in mathematics and related fields. So, let's get started and demystify this process together!

Method 1 Substitution Method

The substitution method is a powerful technique for solving systems of equations. It involves solving one equation for one variable and then substituting that expression into the other equation. This eliminates one variable, allowing us to solve for the remaining variable. Once we have the value of one variable, we can substitute it back into either of the original equations to find the value of the other variable. Let's walk through the steps for our system: $\left\{\begin{aligned}-5 x+y & =3 \\ 10 x-3 y & =-5\end{aligned}\right.$.

Step 1 Isolate a Variable

The first step in the substitution method is to isolate one variable in one of the equations. Looking at our system, it seems easiest to isolate $y$ in the first equation, $-5x + y = 3$. To do this, we simply add $5x$ to both sides of the equation:

$y = 5x + 3$

Now we have $y$ expressed in terms of $x$. This is a crucial step because we can now substitute this expression for $y$ into the second equation. Isolating the variable strategically can make the subsequent steps much simpler. For instance, if an equation already has a variable with a coefficient of 1, that's usually the best one to isolate. In our case, the $y$ in the first equation was perfect. Always keep an eye out for these opportunities, as they can save you time and effort. Remember, the goal is to simplify the system and reduce it to a single equation with one variable. This makes the problem much more manageable. By isolating $y$, we've set the stage for the next step, where we'll use this expression to eliminate $y$ from the second equation.

Step 2 Substitute

Now that we have $y = 5x + 3$, we can substitute this expression into the second equation, $10x - 3y = -5$. This means wherever we see $y$ in the second equation, we'll replace it with $5x + 3$. Here's how it looks:

$10x - 3(5x + 3) = -5$

This substitution is the heart of the method. By replacing $y$ with an expression involving $x$, we've transformed the second equation into an equation with only one variable, $x$. This is a significant step because we now have a single equation that we can solve directly for $x$. It's important to be careful with the substitution, especially when there are coefficients or negative signs involved. Make sure you distribute correctly and maintain the integrity of the equation. The purpose of substitution is to eliminate one variable, making the system solvable. We've successfully done that here, and we're one step closer to finding the solution. Now, the next step is to simplify and solve this equation for $x$. Remember, our goal is to find the value of $x$ that satisfies both equations in the system. By substituting, we've created a pathway to find that value.

Step 3 Solve for x

With our substituted equation, $10x - 3(5x + 3) = -5$, it's time to solve for $x$. First, we need to distribute the $-3$ across the terms inside the parentheses:

$10x - 15x - 9 = -5$

Next, combine like terms on the left side of the equation:

$-5x - 9 = -5$

Now, add 9 to both sides to isolate the term with $x$:

$-5x = 4$

Finally, divide both sides by $-5$ to solve for $x$:

$x = -\frac{4}{5}$

So, we've found our value for $x$! This is a major milestone. Solving for $x$ involves using basic algebraic principles like distribution, combining like terms, and isolating the variable. Each of these steps is crucial to arriving at the correct value. After distributing and combining like terms, we simplified the equation to a more manageable form. Then, using inverse operations, we isolated $x$ and found its value. It's important to double-check your work at each step to avoid making errors. A small mistake in the arithmetic can lead to an incorrect solution. Now that we have the value of $x$, we're halfway to solving the system. The next step is to substitute this value back into one of the original equations (or the expression we found for $y$) to find the value of $y$.

Step 4 Substitute x to find y

Now that we know $x = -\frac{4}{5}$, we can substitute this value back into the equation we found for $y$ in Step 1, which was $y = 5x + 3$. Here's how we do it:

$y = 5(-\frac{4}{5}) + 3$

First, multiply $5$ by $-\frac{4}{5}$:

$y = -4 + 3$

Then, add $-4$ and $3$:

$y = -1$

So, we've found that $y = -1$. This is the final piece of the puzzle! Substituting the value of $x$ to find $y$ is a straightforward process, but it's crucial to be accurate with your arithmetic. Using the previously found value of $x$, we plugged it into the expression for $y$ that we derived earlier. This allowed us to directly calculate the value of $y$. It's important to remember that we could have substituted the value of $x$ into either of the original equations as well. However, using the expression we already had for $y$ simplified the calculation in this case. Always choose the equation or expression that seems easiest to work with to minimize the chances of making a mistake. Now that we have both $x$ and $y$, we can write the solution as an ordered pair. And, of course, we should always check our solution to make sure it's correct.

Step 5 Write the Solution

We've found that $x = -\frac{4}{5}$ and $y = -1$. Therefore, the solution to the system of equations is the ordered pair $\left(-\frac{4}{5}, -1\right)$. This ordered pair represents the point where the two lines represented by our equations intersect on a graph. It's the unique pair of values that satisfies both equations simultaneously. Writing the solution as an ordered pair is the standard way to represent the solution to a system of two equations in two variables. The first value in the pair is always the $x$-coordinate, and the second value is the $y$-coordinate. This notation provides a clear and concise way to communicate the solution. Remember, the solution to a system of equations is not just a set of numbers; it's a point in the coordinate plane. It's the intersection point of the lines represented by the equations. So, by writing the solution as an ordered pair, we're providing a complete and meaningful answer. And as always, let's not forget to check our solution to ensure it's correct!

Step 6 Check the Solution

To make sure our solution $\left(-\frac{4}{5}, -1\right)$ is correct, we need to substitute these values back into both of the original equations and see if they hold true. Let's start with the first equation, $-5x + y = 3$:

$-5(-\frac{4}{5}) + (-1) = 3$

Simplify:

$4 - 1 = 3$

$3 = 3$ (This is true!)

Now, let's check the second equation, $10x - 3y = -5$:

$10(-\frac{4}{5}) - 3(-1) = -5$

Simplify:

$-8 + 3 = -5$

$-5 = -5$ (This is also true!)

Since our solution satisfies both equations, we can confidently say that $\left(-\frac{4}{5}, -1\right)$ is the correct solution to the system. Checking the solution is a crucial step in solving any system of equations. It's your way of verifying that you haven't made any mistakes along the way. By substituting the values of $x$ and $y$ back into the original equations, you can confirm that they make both equations true. If the solution doesn't satisfy both equations, then you know you need to go back and check your work. Checking is particularly important when dealing with fractions or negative signs, as these are common areas for errors. So, always take the time to check your solution. It's a quick way to ensure accuracy and build confidence in your answer.

Method 2 Elimination Method

The elimination method, also known as the addition method, is another powerful technique for solving systems of equations. Instead of solving for one variable and substituting, this method focuses on eliminating one of the variables by adding the equations together. To do this, we often need to multiply one or both equations by a constant so that the coefficients of one of the variables are opposites. This way, when we add the equations, that variable cancels out, leaving us with a single equation in one variable. Let's apply this method to our system: $\left\{\begin{aligned}-5 x+y & =3 \\ 10 x-3 y & =-5\end{aligned}\right.$.

Step 1 Multiply Equations (if needed)

The goal of this step is to make the coefficients of either $x$ or $y$ opposites in the two equations. Looking at our system, $\left\{\begin{aligned}-5 x+y & =3 \\ 10 x-3 y & =-5\end{aligned}\right.$, we see that the coefficients of $x$ are $-5$ and $10$. Notice that if we multiply the first equation by $2$, the coefficient of $x$ will become $-10$, which is the opposite of $10$ in the second equation. So, let's multiply the entire first equation by $2$:

$2(-5x + y) = 2(3)$

This gives us:

$-10x + 2y = 6$

Now our system looks like this:

$\left\{\begin{aligned}-10x + 2y & = 6 \\ 10 x-3 y & =-5\end{aligned}\right.$

Multiplying equations by a constant is a key step in the elimination method. The purpose is to create matching coefficients (with opposite signs) for one of the variables. This sets us up to eliminate that variable in the next step. When choosing which equation to multiply and by what constant, look for the easiest way to create opposites. Sometimes you'll need to multiply both equations, but in this case, multiplying just the first equation by $2$ was sufficient. Remember to multiply every term in the equation, including the constant term on the right side. A common mistake is to forget to multiply the constant, which can lead to an incorrect solution. Once you've multiplied the equation(s), double-check your work to ensure accuracy. With the coefficients of $x$ now opposites, we're ready to add the equations together.

Step 2 Add the Equations

Now that we have the system $\left\{\begin{aligned}-10x + 2y & = 6 \\ 10 x-3 y & =-5\end{aligned}\right.$, we can add the two equations together. When we add the equations, we add the corresponding terms. The $-10x$ and $10x$ terms cancel out, which is exactly what we wanted!

(-10x + 2y) + (10x - 3y) = 6 + (-5)

This simplifies to:

$-y = 1$

Adding the equations is the core of the elimination method. By carefully choosing our multipliers in the previous step, we've ensured that one of the variables cancels out when we add the equations. This leaves us with a single equation in one variable, which we can easily solve. It's crucial to line up the like terms correctly before adding. This means adding the $x$ terms together, the $y$ terms together, and the constant terms together. Adding the equations vertically makes this process easier and helps to avoid mistakes. Remember, the goal is to eliminate one variable, simplifying the system. In this case, the $x$ terms canceled out, leaving us with an equation involving only $y$. This is a major step forward in solving the system. Now, we just need to solve the resulting equation for $y$.

Step 3 Solve for y

We have the equation $-y = 1$. To solve for $y$, we simply multiply both sides by $-1$:

$y = -1$

So, we've found the value of $y$! Solving for $y$ in this step is straightforward. We had a simple equation, $-y = 1$, and a single operation (multiplying by $-1$) was all it took to isolate $y$. It's important to pay attention to the signs when solving equations. A common mistake is to forget to multiply the constant term on the right side by $-1$, which would lead to an incorrect value for $y$. Once you have the value of one variable, you're one step closer to solving the system. In this case, we've found $y$, and now we need to find $x$. We can do this by substituting the value of $y$ back into one of the original equations. Finding the value of $y$ is a significant achievement. It means we've successfully eliminated $x$ and isolated one of the variables. This is a testament to the power of the elimination method.

Step 4 Substitute y to find x

Now that we know $y = -1$, we can substitute this value into either of the original equations to find $x$. Let's use the first equation, $-5x + y = 3$:

$-5x + (-1) = 3$

Add 1 to both sides:

$-5x = 4$

Divide both sides by $-5$:

$x = -\frac{4}{5}$

So, we've found that $x = -\frac{4}{5}$. Substituting the value of $y$ to find $x$ is a familiar step. We've done something similar in the substitution method. The key is to choose an equation that looks easiest to work with. In this case, the first equation seemed like a good choice. Remember to substitute the value carefully, paying attention to the signs. Once you've substituted, it's just a matter of solving the resulting equation for the remaining variable. In this case, we added 1 to both sides and then divided by $-5$ to isolate $x$. With both $x$ and $y$ found, we're almost done! We just need to write the solution as an ordered pair and, of course, check our answer to make sure it's correct.

Step 5 Write the Solution

We've found that $x = -\frac{4}{5}$ and $y = -1$. Therefore, the solution to the system of equations is the ordered pair $\left(-\frac{4}{5}, -1\right)$. This ordered pair represents the point where the two lines represented by our equations intersect. It's the unique pair of values that satisfies both equations simultaneously. Writing the solution as an ordered pair is the standard way to represent the solution to a system of two equations in two variables. The first value in the pair is always the $x$-coordinate, and the second value is the $y$-coordinate. This notation provides a clear and concise way to communicate the solution. Remember, the solution to a system of equations is not just a set of numbers; it's a point in the coordinate plane. It's the intersection point of the lines represented by the equations. So, by writing the solution as an ordered pair, we're providing a complete and meaningful answer. And as always, let's not forget to check our solution to ensure it's correct!

Step 6 Check the Solution

To make sure our solution $\left(-\frac{4}{5}, -1\right)$ is correct, we need to substitute these values back into both of the original equations and see if they hold true. Let's start with the first equation, $-5x + y = 3$:

$-5(-\frac{4}{5}) + (-1) = 3$

Simplify:

$4 - 1 = 3$

$3 = 3$ (This is true!)

Now, let's check the second equation, $10x - 3y = -5$:

$10(-\frac{4}{5}) - 3(-1) = -5$

Simplify:

$-8 + 3 = -5$

$-5 = -5$ (This is also true!)

Since our solution satisfies both equations, we can confidently say that $\left(-\frac{4}{5}, -1\right)$ is the correct solution to the system. Checking the solution is a crucial step in solving any system of equations. It's your way of verifying that you haven't made any mistakes along the way. By substituting the values of $x$ and $y$ back into the original equations, you can confirm that they make both equations true. If the solution doesn't satisfy both equations, then you know you need to go back and check your work. Checking is particularly important when dealing with fractions or negative signs, as these are common areas for errors. So, always take the time to check your solution. It's a quick way to ensure accuracy and build confidence in your answer.

Final Answer

So, after working through both the substitution and elimination methods, we've arrived at the same solution: the solution to the system $\left\{\begin{aligned}-5 x+y & =3 \\ 10 x-3 y & =-5\end{aligned}\right.$ is $(x, y) = \left(-\frac{4}{5}, -1\right)$. You've now seen two different approaches to solving systems of equations, and you can choose the method that you find most comfortable or efficient. Keep practicing, and you'll become a system-solving master in no time! Remember, mathematics is all about practice, so the more you work on these problems, the easier they will become. And don't forget to always check your answers! This will help you catch any mistakes and build your confidence. Great job, everyone!