Solving Equations: A Step-by-Step Guide

Hey there, math enthusiasts! Today, we're diving into the world of solving systems of equations using the substitution method. Don't worry, it sounds more complicated than it is! We'll break down the process step by step, making it super easy to understand. So, grab your pencils and let's get started.

We will be solving the following system of equations:

$\frac{3}{8} x+\frac{1}{3} y=\frac{17}{24}$

$x+7 y=8$

Understanding the Substitution Method

Before we jump into the problem, let's quickly recap what the substitution method is all about. The substitution method is a cool technique used to solve systems of equations. It's like a mathematical detective game where we aim to find the values of the variables that satisfy all the equations in the system. The main idea behind substitution is to solve one of the equations for one variable and then substitute that expression into the other equation. This transforms the problem into a single equation with only one variable, which we can then solve. Once we have the value of that variable, we can substitute it back into one of the original equations to find the value of the other variable. Voila! We've solved the system!

This method is particularly handy when one of the equations is already solved for a variable or can be easily rearranged to isolate a variable. It's a fundamental skill in algebra and is used extensively in various fields like physics, engineering, and economics. Mastering the substitution method opens doors to solving more complex problems and understanding how different variables interact within a system. We'll be using this method in solving this problem, step by step, so that you won't get lost along the way. Get ready to flex those equation-solving muscles!

For example, if you have two equations, you first solve one of the equations for either x or y. Then, you substitute the expression you found for that variable into the other equation. You now have one equation with one variable, which you solve. Once you know that variable's value, substitute it back into either original equation to find the value of the other variable. And just like that, you've solved the system! You will find this a lot easier after we solve our example problem.

Step-by-Step Solution

Now, let's get down to the nitty-gritty and solve our system of equations using the substitution method. We'll walk through each step, making sure you grasp every detail.

Step 1: Isolate a Variable

First, we need to choose one of the equations and solve it for one of the variables. Looking at our system, it's easiest to solve the second equation, $x+7y=8$, for $x$. This is because $x$ already has a coefficient of 1, making the isolation process straightforward. Let's rearrange the second equation to solve for $x$:

$x + 7y = 8$

Subtract $7y$ from both sides:

$x = 8 - 7y$

Now we have an expression for $x$ in terms of $y$.

Step 2: Substitute the Expression

Next, we'll substitute the expression we found for $x$ (which is $8 - 7y$) into the other equation. The other equation is $\frac{3}{8} x+\frac{1}{3} y=\frac{17}{24}$. Let's replace $x$ in this equation with $(8 - 7y)$:

$\frac{3}{8} (8 - 7y) + \frac{1}{3} y = \frac{17}{24}$

Step 3: Simplify and Solve for the Remaining Variable

Now we have a single equation with only one variable, $y$. Let's simplify and solve for $y$. First, distribute the $\frac{3}{8}$:

$\frac{3}{8} * 8 - \frac{3}{8} * 7y + \frac{1}{3} y = \frac{17}{24}$

$3 - \frac{21}{8}y + \frac{1}{3}y = \frac{17}{24}$

Next, let's combine the $y$ terms. To do this, we need a common denominator, which is 24. So, we'll rewrite $\frac{21}{8}y$ and $\frac{1}{3}y$ with a denominator of 24:

$\frac{21}{8}y = \frac{21 * 3}{8 * 3}y = \frac{63}{24}y$

$\frac{1}{3}y = \frac{1 * 8}{3 * 8}y = \frac{8}{24}y$

Now our equation looks like this:

$3 - \frac{63}{24}y + \frac{8}{24}y = \frac{17}{24}$

Combine the $y$ terms:

$3 - \frac{55}{24}y = \frac{17}{24}$

Subtract 3 (or $\frac{72}{24}$) from both sides:

$- \frac{55}{24}y = \frac{17}{24} - \frac{72}{24}$

$- \frac{55}{24}y = -\frac{55}{24}$

Finally, divide both sides by $-\frac{55}{24}$. This is the same as multiplying by $-\frac{24}{55}$:

$y = -\frac{55}{24} * -\frac{24}{55}$

$y = 1$

We've found that $y = 1$!

Step 4: Substitute to Find the Other Variable

Now that we know $y = 1$, we can substitute this value back into one of our original equations to find $x$. Remember, we had $x = 8 - 7y$. Let's substitute $y = 1$ into this equation:

$x = 8 - 7(1)$

$x = 8 - 7$

$x = 1$

So, we've found that $x = 1$.

Step 5: State the Solution

Therefore, the solution to the system of equations is $x = 1$ and $y = 1$. We can write this as an ordered pair: $(1, 1)$. This represents the point where the two lines represented by our equations intersect on a graph. This is the solution to the system of equations.

Verifying Your Solution

It's always a good idea to check your solution to make sure it's correct. We can do this by substituting the values of $x$ and $y$ back into both original equations and verifying that both equations are true. Let's plug $x=1$ and $y=1$ into the first equation:

$\frac{3}{8}(1) + \frac{1}{3}(1) = \frac{17}{24}$

$\frac{3}{8} + \frac{1}{3} = \frac{17}{24}$

To add the fractions, find a common denominator (24):

$\frac{9}{24} + \frac{8}{24} = \frac{17}{24}$

$\frac{17}{24} = \frac{17}{24}$

The first equation holds true!

Now, let's plug $x=1$ and $y=1$ into the second equation:

$1 + 7(1) = 8$

$1 + 7 = 8$

$8 = 8$

The second equation also holds true! Since both equations are true, we know our solution $(1, 1)$ is correct. This is the correct solution to the system of equations. Always check your answers to catch any errors and build confidence in your problem-solving skills.

Tips for Success

Here are some handy tips to help you master the substitution method:

• Choose Wisely: When isolating a variable, pick the equation and variable that look easiest to solve for. This will often be the variable with a coefficient of 1 or -1.
• Be Careful with Signs: Pay close attention to positive and negative signs. A small mistake here can lead to a wrong answer.
• Simplify, Simplify, Simplify: Always simplify your equations as much as possible to avoid unnecessary complications.
• Double-Check Your Work: After finding your solution, always plug it back into both original equations to verify that it's correct. This can catch any errors you might have made along the way.
• Practice Makes Perfect: The more you practice, the better you'll become at solving systems of equations using substitution. Try different examples and challenge yourself with more complex problems.

Conclusion

And there you have it! You've successfully solved a system of equations using the substitution method. By following these steps and practicing regularly, you'll become a pro at solving these types of problems. Remember, the key is to understand each step and to practice consistently. Keep up the great work, and happy solving, guys! You got this! This method is a fundamental skill in algebra, so keep practicing. Now go forth and conquer those equations! Hopefully, this guide helped you a lot!