Solving Systems: Find 'y' In Equations!

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Hey math enthusiasts! Today, we're diving into a classic algebra problem: solving a system of equations. Specifically, we're going to figure out the value of y in a given system. Don't worry if it sounds intimidating; we'll break it down step by step, making it super easy to understand. So, grab your pens and paper, and let's get started! Our main goal is to find the solution for the value of y. This is a common problem type in mathematics, and it's super important for understanding more advanced concepts down the line. We'll utilize the techniques of solving systems of linear equations using the substitution method or the elimination method. By the end of this article, you will be able to solve similar problems with confidence. The ability to solve these kinds of equations is essential for success in math, and in many real-world applications. Let's look at the given system of equations:

13x+14y=12x−3y=−30\begin{array}{l} \frac{1}{3} x+\frac{1}{4} y=1 \\ 2 x-3 y=-30 \end{array}

This is a classic example of a system of two linear equations with two variables, x and y. Our mission? To find the values of x and y that satisfy both equations simultaneously. There are a couple of popular methods we can use: substitution and elimination. I'll walk you through one of these methods. No matter which method you choose, the key is to be systematic and careful with your calculations. Make sure to double-check your work along the way to avoid any silly mistakes. Remember, practice makes perfect. The more problems you solve, the more comfortable you'll become with these techniques. Now, let's go over the first step, understanding the problem. Understanding the problem is the first step in solving any mathematical problem. Identify what is given and what you need to find. In this case, we have two equations, and we want to find the values of x and y that satisfy both equations. Once we understand what's being asked, we can start thinking about how to solve it. It is essential to go step by step, which is a great approach for solving equations. That way, we can avoid any errors and make sure we get the correct solution.

Method: Solving for y using the Elimination Method

Alright, guys, let's tackle this problem using the elimination method! This is a great choice because it lets us cancel out one of the variables. Now, before we start, let's talk about the elimination method, and what it really means. The elimination method involves manipulating the equations so that when you add or subtract them, one of the variables disappears (is eliminated). This leaves us with a single equation and a single variable, which we can then easily solve. It's like a clever trick to simplify the problem! The main concept behind the elimination method is to manipulate the equations in a way that allows us to eliminate one of the variables when we add or subtract the equations. To do this effectively, we need to make sure the coefficients of either x or y are opposites. That means they should have the same absolute value, but opposite signs. This will allow us to cancel them out when we add the equations. So, here's how we'll do it step-by-step:

  1. Prepare the Equations: We need to get the coefficients of either x or y to be opposites. Let's work on getting the x coefficients to be opposites. Multiply the first equation by 6: This will help us eliminate the x variable. Why 6? Because it's the least common multiple (LCM) of 3 and 2. Multiplying by 6 will give us coefficients of 2 and 12 for the x terms, which are not opposites. To get opposites, let's manipulate the equations:

    • Multiply the first equation by 2: 2 * (1/3x + 1/4y = 1) becomes (2/3)x + (1/2)y = 2. Let's rewrite the second equation to make it easier to work with. Multiply the second equation by 1: 1 * (2x - 3y = -30) remains 2x - 3y = -30. Now, let's try to eliminate the x variable by multiplying the first equation by -3: -3 * (1/3x + 1/4y = 1) becomes -x - (3/4)y = -3. Let's rewrite the second equation to make it easier to work with. Multiply the second equation by 1: 1 * (2x - 3y = -30) remains 2x - 3y = -30. Now, we're ready for the next step, which is combining the equations. Our goal is to manipulate the equations so that when we add them together, one of the variables is eliminated. This makes the solving process easier and reduces the chances of errors. It's like finding the perfect combination of numbers to cancel each other out. Now, let's move on to the next step, which will bring us closer to the solution. Be patient, and don't worry, we're getting closer to solving the problem. The most important thing is to understand the steps involved and to double-check your work along the way. Stay focused, and we'll have our answer soon!

  2. Eliminate x: To eliminate x, we can multiply the first equation by -2 and keep the second equation as it is. This will give us opposite coefficients for the x terms. So, we'll have:

    • -2 * (1/3x + 1/4y = 1) becomes (-2/3)x - (1/2)y = -2
    • 2x - 3y = -30 remains 2x - 3y = -30

  3. Add the Equations: Now, add the modified equations together. This cancels out the x terms:

    • (-2/3)x - (1/2)y = -2
    • 2x - 3y = -30

    The equations do not directly eliminate the x terms. It looks like we're not quite ready to eliminate x directly. Instead, let's focus on eliminating x in the following way. Multiply the first equation by 2:

    • 2 * (1/3x + 1/4y = 1) becomes (2/3)x + (1/2)y = 2
    • 2x - 3y = -30 remains 2x - 3y = -30

    Now, we have (2/3)x + (1/2)y = 2 and 2x - 3y = -30. To eliminate x, we need to multiply the first equation by -3 and the second equation by 1, so the coefficients for the x terms become opposites. So, we'll have:

    • -3 * (1/3x + 1/4y = 1) becomes -x - (3/4)y = -3
    • 2x - 3y = -30 remains 2x - 3y = -30

    To eliminate the x variable, we need to multiply the first equation by -2 and the second equation by 3. Doing so will make the coefficients of x opposites (-2/3 and 6) so they cancel out. The equations will then become:

    • -2 * (1/3x + 1/4y = 1) becomes (-2/3)x - (1/2)y = -2
    • 3 * (2x - 3y = -30) becomes 6x - 9y = -90

    Now, we will have a new system of equations. Add the new equations together: (-2/3)x - (1/2)y = -2 6x - 9y = -90

    Add the equations to eliminate x. The elimination method is a powerful technique for solving systems of equations. By carefully manipulating the equations, we can eliminate one of the variables and solve for the other. This method is especially useful when the coefficients of one of the variables are easily made opposites. The key is to be systematic and organized in your approach, and to double-check your calculations at each step. Now, add the equations together.

  4. Combine the y terms and constants:

    To proceed with the elimination method, we have to manipulate the equations in a way that allows us to eliminate either x or y. Once we've done that, we'll be left with a single equation with a single variable, which is much easier to solve. Let's focus on eliminating x. We have:

    • -2 * (1/3x + 1/4y = 1) becomes (-2/3)x - (1/2)y = -2
    • 3 * (2x - 3y = -30) becomes 6x - 9y = -90
    • Let's add the equations: (-2/3)x - (1/2)y + 6x - 9y = -2 - 90

    After combining like terms we have:

    • (16/3)x - (19/2)y = -92

    Now that we are left with both x and y, we need to take a step back and revisit our previous steps.

    • Let's return to step 1. Prepare the equations: Let's multiply the first equation by 4 and the second equation by 1.

    • 4 * (1/3x + 1/4y = 1) becomes (4/3)x + y = 4

    • 2x - 3y = -30 remains 2x - 3y = -30

    • Multiply the first equation by 3:

    • 3 * ((4/3)x + y = 4) becomes 4x + 3y = 12

    • 2x - 3y = -30 remains 2x - 3y = -30

    • Add the two equations:

    • 4x + 3y + 2x - 3y = 12 - 30

    • 6x = -18

    • x = -3

  5. Solve for y: Now that we know x = -3, we can substitute this value back into either of the original equations to solve for y. Let's use the first equation: (1/3)x + (1/4)y = 1. Substitute x = -3: (1/3)(-3) + (1/4)y = 1, which simplifies to -1 + (1/4)y = 1. Add 1 to both sides: (1/4)y = 2. Multiply both sides by 4: y = 8.

Solution

Therefore, the value of y in the solution to the system of equations is 8. Congrats, you have solved the equation! The elimination method is a useful tool in your mathematical toolkit! Remember, practice makes perfect. Keep solving different types of equations, and you'll become more and more confident in your abilities. Every problem you solve brings you closer to mastering algebra. Keep going, you got this! Let me know if you want to try another problem! Have fun! And that's all, folks!