Solving Systems: Find X And Y With Ease!

Hey Plastik Magazine readers! Ever stumbled upon a system of equations and felt a bit lost? Don't sweat it, because today, we're diving deep into how to solve them, specifically focusing on the following system:

$4y - 5 = 20 - 3x$

$4x - 7y + 16 = 0$

Our mission, should we choose to accept it, is to find the values of x and y that make both equations true. It's like a mathematical treasure hunt, and the prize is the solution (x, y). Let's break it down step by step, making it super clear and easy to follow. Remember, understanding these concepts can seriously boost your math game, whether you're a student or just someone who loves a good mental challenge. We will start by understanding the system of equations. This involves recognizing that we have two equations, each representing a relationship between x and y. The goal is to find the point (or points) where these relationships intersect. In this case, we're looking for a single point, but systems can have no solutions or infinitely many. Think of it like two lines on a graph. The solution is where they meet. The first equation, $4y - 5 = 20 - 3x$, can be rearranged to look like a standard linear equation, making it easier to work with. Likewise, the second equation, $4x - 7y + 16 = 0$, also represents a linear relationship. The key is to manipulate these equations until we can isolate x and y. Now, let's look at preparing the equations for solving. The first step in solving this system is to rearrange the equations into a more manageable form. We can start by rewriting the first equation $4y - 5 = 20 - 3x$. By adding $3x$ to both sides and adding $5$ to both sides, we get $3x + 4y = 25$. This puts it in a standard linear form, where all the variables are on one side and the constant on the other. Next, let's rearrange the second equation, $4x - 7y + 16 = 0$. By subtracting 16 from both sides, we obtain $4x - 7y = -16$. Now, we have two equations that are easier to work with. These steps are crucial because they set the stage for using methods like substitution or elimination, which are the main tools for solving systems of equations. Getting the equations into this form makes the solving process much smoother and less prone to errors.

Solving the System Using Substitution Method

Alright, let's get down to the nitty-gritty and solve the system using the substitution method. This method involves isolating one variable in one of the equations and then substituting that expression into the other equation. Let's begin with our rearranged equations:

$3x + 4y = 25$

$4x - 7y = -16$

From the first equation, we can solve for x: $3x = 25 - 4y$, which gives us $x = (25 - 4y) / 3$. Now that we have an expression for x, we substitute this into the second equation: $4((25 - 4y) / 3) - 7y = -16$. The next step involves simplifying this equation to solve for y. This is the core of the substitution method: replacing one variable with an equivalent expression from another equation to reduce the system to a single-variable equation. Let's simplify and solve for y. We have the equation $4((25 - 4y) / 3) - 7y = -16$. First, multiply both sides by 3 to eliminate the fraction: $4(25 - 4y) - 21y = -48$. Then, distribute the 4: $100 - 16y - 21y = -48$. Combine like terms: $100 - 37y = -48$. Subtract 100 from both sides: $-37y = -148$. Finally, divide by -37: $y = 4$. We found the value of y! Next, we need to find the value of x. Now that we've found y, we can easily find x by substituting y = 4 into either of the original equations. Let's use the rearranged equation $x = (25 - 4y) / 3$. Substituting y = 4, we get $x = (25 - 4*4) / 3$, which simplifies to $x = (25 - 16) / 3$, and further to $x = 9 / 3$. Therefore, $x = 3$. We've done it, guys! We have successfully solved the system of equations. Our solution is (x, y) = (3, 4).

Solving the System Using the Elimination Method

Alternatively, let's solve the system using the elimination method. This approach involves manipulating the equations so that when you add or subtract them, one of the variables is eliminated. It's all about making the coefficients of either x or y opposites. Let's revisit our system of equations:

$3x + 4y = 25$

$4x - 7y = -16$

To eliminate x, we can multiply the first equation by 4 and the second equation by -3. This gives us: $12x + 16y = 100$ and $-12x + 21y = 48$. Notice how the coefficients of x are now opposites? Adding these two equations will eliminate x. Next, let's add the equations together. Adding the modified equations, $12x + 16y = 100$ and $-12x + 21y = 48$, we get $37y = 148$. This simplifies to $y = 4$. Excellent, we have found y! Now, to find x, we substitute y = 4 into either of the original equations. Let's use the first equation: $3x + 4(4) = 25$, which simplifies to $3x + 16 = 25$. Subtracting 16 from both sides, we get $3x = 9$, and dividing by 3, we find $x = 3$. Voila! The solution, again, is (x, y) = (3, 4). This confirms our previous result, showing the versatility of different methods.

Checking Your Solution

It's always a smart move to verify the solution. Before we wrap up, let's quickly check our solution to make sure it's correct. We found that x = 3 and y = 4. Let's plug these values into both of the original equations to confirm:

$4y - 5 = 20 - 3x$ becomes $4(4) - 5 = 20 - 3(3)$, which simplifies to $16 - 5 = 20 - 9$, and further to $11 = 11$. This checks out! $4x - 7y + 16 = 0$ becomes $4(3) - 7(4) + 16 = 0$, which simplifies to $12 - 28 + 16 = 0$, and further to $0 = 0$. Also, checks out! Since both equations are true with these values, we know that our solution (3, 4) is correct.

Conclusion: Mastering Systems of Equations

Alright, folks, that wraps up our exploration of solving systems of equations! We've tackled it step-by-step, from rearranging the equations to using both substitution and elimination methods. Remember, practice is key. The more you work with these types of problems, the easier and more intuitive they'll become. Whether you're a math enthusiast, a student, or just someone who enjoys a good puzzle, understanding how to solve systems of equations is a valuable skill. It's applicable in many areas, from everyday problem-solving to advanced scientific and engineering applications. So, keep practicing, and don't be afraid to experiment with different methods. You've got this!