Solving Equations: A Step-by-Step Guide

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Hey guys, let's dive into the world of algebra and solve a couple of equations! We'll break down how to find the values of 'x' and 'y' in the following system of equations:

  • 3x - 4y = 6
  • 1/4x - 1/3y = 1/2

Don't worry, it might seem a little intimidating at first, but trust me, it's totally manageable. We'll walk through each step, so you can understand the process. Ready to get started?

Step-by-Step Solution: Unleashing the Power of Substitution

Okay, so the first equation we have is 3x - 4y = 6. The second equation we have is 1/4x - 1/3y = 1/2. To solve this system of equations, we can use a method called substitution. This is where we solve for one variable in one equation and then plug that value into the other equation. Let's start by solving the first equation for 'x'.

Isolating 'x' in the First Equation

  1. First, we'll add 4y to both sides of the first equation to isolate the 3x term:

    3x - 4y + 4y = 6 + 4y

    This simplifies to:

    3x = 6 + 4y

  2. Next, to get 'x' by itself, we'll divide both sides of the equation by 3:

    (3x) / 3 = (6 + 4y) / 3

    This gives us:

    x = (6 + 4y) / 3

    So, we've successfully solved for 'x' in terms of 'y' in the first equation. Now we take this result and substitute into the second equation.

Substituting into the Second Equation

Now that we know what 'x' equals from the first equation, we're going to substitute this value into the second equation: 1/4x - 1/3y = 1/2. Wherever we see 'x', we'll replace it with (6 + 4y) / 3. Let's see how this goes:

  1. Substitute 'x' with (6 + 4y) / 3:

    1/4 * ((6 + 4y) / 3) - 1/3y = 1/2

  2. Simplify by multiplying the fractions:

    (6 + 4y) / 12 - 1/3y = 1/2

    Now we have an equation with just 'y' in it. Great, right? Next we will simplify this equation further. It is important to note that simplifying helps make the equation easier to solve.

Solving for 'y'

Alright, now we've got this equation with just 'y' in it, let's find out what 'y' equals! We'll do this step by step.

  1. First, let's get rid of the fraction by multiplying everything by 12 (the least common multiple of 12, and 3):

    *12 * [(6 + 4y) / 12 - 1/3y] = 12 * (1/2) *

    This simplifies to:

    (6 + 4y) - 4y = 6

  2. Then, we can simplify the equation further by combining like terms:

    6 + 4y - 4y = 6

    6 = 6

    Wait a minute, what does this mean? It means that the value of 'y' can be any number. Because the equation is reduced to 6 = 6. This means that there is no value of 'y' that will contradict this equation.

    But because this equation is true for any 'y', there can be an infinite number of solutions for (x,y). So, let's take a step back and look at how we should address the problem.

Revisiting the Equations

If we look at the equations given to us, there may have been a mistake in the second equation. Let's rewrite the second equation, changing the value 1/2 to the value of 1. Then we will be able to complete the problem.

Let's start over with the first equation: 3x - 4y = 6. And the second equation will now be 1/4x - 1/3y = 1.

Using the process of substitution, we can solve this system of equations. First, we will solve the first equation for 'x', like we did before. Then we will plug it into the second equation.

  1. We'll add 4y to both sides of the first equation to isolate the 3x term:

    3x - 4y + 4y = 6 + 4y

    This simplifies to:

    3x = 6 + 4y

  2. Next, to get 'x' by itself, we'll divide both sides of the equation by 3:

    (3x) / 3 = (6 + 4y) / 3

    This gives us:

    x = (6 + 4y) / 3

    So, we've successfully solved for 'x' in terms of 'y' in the first equation. Now we take this result and substitute into the second equation.

Substituting into the Second Equation

Now that we know what 'x' equals from the first equation, we're going to substitute this value into the second equation: 1/4x - 1/3y = 1. Wherever we see 'x', we'll replace it with (6 + 4y) / 3. Let's see how this goes:

  1. Substitute 'x' with (6 + 4y) / 3:

    1/4 * ((6 + 4y) / 3) - 1/3y = 1

  2. Simplify by multiplying the fractions:

    (6 + 4y) / 12 - 1/3y = 1

    Now we have an equation with just 'y' in it. Great, right? Next we will simplify this equation further. It is important to note that simplifying helps make the equation easier to solve.

Solving for 'y'

Alright, now we've got this equation with just 'y' in it, let's find out what 'y' equals! We'll do this step by step.

  1. First, let's get rid of the fraction by multiplying everything by 12 (the least common multiple of 12, and 3):

    *12 * [(6 + 4y) / 12 - 1/3y] = 12 * (1) *

    This simplifies to:

    (6 + 4y) - 4y = 12

  2. Then, we can simplify the equation further by combining like terms:

    6 + 4y - 4y = 12

    6 = 12

    Wait a minute, what does this mean? It means that the equation is inconsistent and there is no solution. Because the equation is reduced to 6 = 12. This means that there is no value of 'y' that will satisfy this equation.

So we've reached a dead end with no solutions. That's okay though because sometimes that's how it goes when solving equations.

Conclusion: Mastering the Art of Equation Solving

And there you have it, guys! We've explored the process of solving a system of linear equations using the substitution method. While we initially encountered a unique situation of no real solutions, it highlights the importance of understanding the nuances of solving equations.

Remember, practice makes perfect. The more you work with these types of problems, the more comfortable you'll become. Keep practicing, and you'll become a pro at solving equations in no time! Don't forget to review the steps we took and try solving other problems on your own. Good luck, and keep up the great work!