Solving Systems Of Equations By Substitution: A Step-by-Step Guide

Hey math enthusiasts! Ever found yourself staring at a pair of equations, feeling a bit lost? Don't sweat it! We're diving into a super useful technique called substitution, a fantastic way to crack the code and find the solutions to systems of equations. In this article, we'll walk through the process step-by-step, making sure you grasp every concept. We'll be using the provided equations: $\frac{3}{8} x+\frac{1}{3} y=\frac{17}{24}$ and $x+7 y=8$. So, grab your pencils and let's get started!

Understanding the Basics: What is a System of Equations?

First things first, what exactly is a system of equations? Basically, it's a set of two or more equations, and we're aiming to find the values of the variables (usually x and y) that satisfy all equations in the system. Think of it like a puzzle where you need to find the pieces that fit perfectly in multiple spots. The solution to a system of equations is the point (or points) where the graphs of the equations intersect. When we have two linear equations, as we do in this case, the solution will usually be a single point, where the two lines cross each other on a graph. This point gives us the specific x and y values that make both equations true at the same time. The essence of solving systems of equations lies in finding those common values. We can solve systems of equations graphically, by graphing each line and finding their intersection, but this is sometimes not very accurate. That's why algebraic methods, like substitution, are super handy because they give us exact solutions.

Here’s a quick analogy: Imagine you have two clues to solve a mystery. Each equation is like a clue, and the solution is the ultimate answer that satisfies both. Our goal is to use the clues (equations) to discover the hidden values of x and y. We're trying to figure out what values work simultaneously in both equations. The substitution method is a cool algebraic technique to find the exact solutions for the variables, giving us a precise answer rather than a visual estimation from a graph. The beauty of this method is its ability to handle complex equations, ensuring accuracy and precision in our solutions. So, when faced with a system of equations, remember the goal: Find the x and y values that make all equations true!

Step 1: Isolate a Variable

Alright, let's get down to business! The first step in the substitution method is to isolate one of the variables in either of the equations. "Isolate" means to get the variable by itself on one side of the equation. Choose the equation that looks easiest to work with. In our case, the second equation, $x + 7y = 8$, is a good starting point because it has a single x without any coefficients (other than 1), which simplifies things. Let's solve the equation for x. We want to rearrange the equation so that x is alone on one side. We'll subtract $7y$ from both sides of the equation. This gives us: $x = 8 - 7y$.

See? We've successfully isolated x! Now we know that x is equal to $8 - 7y$. Keep this in mind, as it's the key to the next steps. Why is isolating a variable so important? Because it gives us a direct expression for that variable in terms of the other. This expression is going to be "substituted" into the other equation, which helps us to reduce the equations down to a single variable equation, which we can easily solve. This is the core concept of the substitution method: using the expression of one variable to replace the same variable in the other equation. By doing this, we create an equation with only one variable, which simplifies our problem considerably. By the end of this step, we've transformed one of the equations into an expression ready for substitution, which moves us closer to finding our solution!

Step 2: Substitute the Expression

This is where the magic happens, guys! Now that we have an expression for x (which is $x = 8 - 7y$), we're going to substitute it into the other equation, which is $\frac{3}{8} x+\frac{1}{3} y=\frac{17}{24}$. Wherever you see x in this equation, replace it with $8 - 7y$. So, the equation $\frac{3}{8} x+\frac{1}{3} y=\frac{17}{24}$ becomes:

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

See how we've replaced x with its equivalent expression? Now, we have an equation with only one variable, y. This is a huge step forward, because it allows us to solve for y. What does this mean? It means we're trading an equation with two variables for one with a single variable, which we can solve. The key is to replace the variable with an equivalent value, simplifying the problem to a solvable equation. The act of substitution is all about finding a relationship between two variables, expressing one in terms of the other, which will simplify our work. By replacing x with the expression we found in Step 1, we transform the original equation into a new one. The new equation has only one variable, y, making it much easier to solve. The aim of this step is to transform our system of two equations into a single equation with one variable, ready for the next phase: solving for y!

Step 3: Solve for the Remaining Variable

Let's roll up our sleeves and solve for y! We have the equation $\frac{3}{8} (8 - 7y) + \frac{1}{3} y = \frac{17}{24}$. First, we need to simplify this equation. Start by distributing the $\frac{3}{8}$: $\frac{3}{8} * 8 = 3$ and $\frac{3}{8} * -7y = -\frac{21}{8} y$. Thus, our equation becomes:

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

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

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

Now, combine the y terms: $-\frac{63}{24} y + \frac{8}{24} y = -\frac{55}{24} y$. Our equation is now:

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

Subtract 3 from both sides:

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

To subtract 3, convert it to a fraction with a denominator of 24: $3 = \frac{72}{24}$. So, the equation becomes:

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

Simplify the right side: $- \frac{55}{24} y = -\frac{55}{24}$.

Finally, to solve for y, divide both sides by $-\frac{55}{24}$. This is the same as multiplying by $-\frac{24}{55}$. So, we get:

$y = -\frac{55}{24} * -\frac{24}{55} = 1$

We did it! We solved for y, and we found that $y = 1$. The hard part is over, guys! This step is all about simplifying and isolating y to figure out its value. By simplifying and manipulating the equation, we bring y closer to the solution. This means isolating y through distributing, combining like terms, and then performing algebraic operations on both sides of the equation. The objective is to unravel the equation, layer by layer, until only y remains on one side, thus revealing the value of y. Remember to always keep your equation balanced; whatever you do to one side, you must do to the other!

Step 4: Substitute Back to Find the Other Variable

We're in the home stretch! We now know that y = 1. Remember the equation we found in Step 1, $x = 8 - 7y$? Now, we can substitute the value of y (which is 1) into this equation to find x. So, we get:

$x = 8 - 7(1)$

$x = 8 - 7$

$x = 1$

Ta-da! We've found that x = 1. So, the solution to the system of equations is x = 1 and y = 1. This means the point where the two lines intersect is (1, 1). To get the other variable, we take the result we got from the last step and put it back into the equation where the variable was already isolated, making it easy to calculate. When the value of one variable is known, it can be substituted into either of the original equations. This substitution unlocks the value of the remaining variable. By doing this substitution, we can calculate the exact value of the remaining variable in the original equations. This gives us the complete set of values which is the solution to our system of equations.

Step 5: Check Your Answer

Always a good idea to check your work, right? To make sure our solution is correct, we'll substitute the values of x = 1 and y = 1 into both original equations. Let's start with the first equation, $\frac{3}{8} x+\frac{1}{3} y=\frac{17}{24}$:

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

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

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

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

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

Great! This equation checks out. Now, let's check the second equation, $x + 7y = 8$:

$1 + 7(1) = 8$

$1 + 7 = 8$

$8 = 8$

This equation also checks out. Since both equations are true with x = 1 and y = 1, we can be confident that our solution is correct! This confirmation step is not just about showing the math is accurate, but also about providing peace of mind. By doing a quick check, we make sure our solution actually works in the original equations. The main goal here is to confirm the values we calculated fit the original equations. Checking our solution by substituting the values into the original equations confirms that our solution satisfies all conditions. Checking your solution is like double-checking your work and making sure everything aligns! So, never skip this step; it provides assurance that your answer is correct and complete.

Conclusion: You've Mastered Substitution!

Awesome work, guys! You've successfully navigated the substitution method and found the solution to a system of equations. Remember, the key steps are to isolate a variable, substitute the expression, solve for the remaining variable, and then substitute back to find the other variable. Always check your answer to ensure accuracy. Practice makes perfect, so keep practicing with different systems of equations, and you'll become a substitution pro in no time! Keep practicing and trying different kinds of equations. Remember, with practice, these methods will become second nature, giving you a powerful tool for solving mathematical problems. Now go forth and conquer those equations! Keep in mind, the more you use these techniques, the more confident you will become in your math abilities! Keep exploring the world of math, and you will discover it is full of fascinating topics and challenges. Well done! You are ready to tackle many more challenging problems!