Solving System Of Equations By Substitution The Step By Step Guide
Hey guys! Let's dive into solving systems of equations using a nifty algebraic technique called substitution. This method is super helpful when you've got one equation that's already solved for one variable, making it easy to plug that expression into another equation. We'll walk through an example step by step, making sure you've got a solid grasp of the process. So, let's get started and make math a little less mysterious!
Understanding Substitution
Before we jump into the nitty-gritty, let’s break down what substitution really means in the context of solving systems of equations. Think of it like this: you're replacing one thing with another that's equal to it. In our case, we're replacing a variable in one equation with an expression from another equation. This clever trick allows us to reduce a system of two equations with two variables into a single equation with just one variable, which is way easier to solve. Once we find the value of that one variable, we can then substitute it back into one of the original equations to find the value of the other variable. Cool, right?
Now, why would we use substitution? Well, it's particularly useful when one of the equations is already solved (or easily solvable) for one variable. For example, if you have an equation like y = 2x + 1, substitution is your best friend. You can simply take that 2x + 1 and replace every y you see in the other equation. This method is also super versatile and can handle both linear and non-linear systems, making it a valuable tool in your mathematical arsenal. Remember, the goal is to simplify the problem, and substitution does just that by eliminating one variable at a time. By the end of this article, you'll be confidently using substitution to tackle a variety of equation systems!
When to Use Substitution
Knowing when to use substitution is just as important as knowing how to use it. Substitution shines when one of your equations is already solved for a variable, or when it's super easy to isolate a variable. For example, if you have an equation like y = 3x - 2, or even something like x + y = 5 (which can be easily rearranged to y = 5 - x), substitution is going to be your go-to method. On the flip side, if you have equations where the coefficients of the variables are neatly lined up, and no variable is easily isolated, you might find that elimination is a more straightforward approach.
Another scenario where substitution excels is when you're dealing with non-linear systems, like the one we're about to solve. These systems involve equations with squared terms, square roots, or other non-linear expressions. Substitution allows you to navigate these complexities by simplifying the equations step by step. In general, look for equations where a variable is expressed in terms of the other – that's your cue to use substitution. So, keep an eye out for those opportunities, and you'll become a substitution pro in no time! Think of it like choosing the right tool for the job; substitution is the perfect tool when you need to unravel equations where one variable is already conveniently expressed in terms of another.
Our System of Equations
Okay, let's get down to the specific system of equations we're going to tackle. We have two equations:
x² + y² = 32y = x
Notice anything special about these equations? The first one is a circle equation (though you don't necessarily need to know that to solve it), and the second one is a simple line. This system is a classic example where substitution is the perfect method. Why? Because the second equation, y = x, is already solved for y. This makes our lives so much easier because we can directly substitute x for y in the first equation.
Before we jump into the substitution, let's take a moment to appreciate what we're about to do. We're essentially going to replace the y in the first equation with x, which will give us an equation with only one variable: x. This is the magic of substitution: it simplifies the problem by reducing the number of variables. We're told that one solution is (-4, -4). That means that when x = -4, y = -4 satisfies both equations. Our goal now is to find the other solution. So, with our equations in hand and our strategy in mind, let's move on to the actual substitution process!
Performing the Substitution
Alright, let's get our hands dirty and actually perform the substitution. Remember, our system of equations is:
x² + y² = 32y = x
The second equation, y = x, is our golden ticket. It tells us that wherever we see a y, we can replace it with an x. So, let's take the first equation, x² + y² = 32, and do just that. We're going to substitute x for y, which gives us:
x² + (x)² = 32
Notice how we've replaced y² with (x)². This is the heart of the substitution method. We've now transformed our equation from having two variables (x and y) to having just one variable (x). This is a huge step because we can now solve for x. The next step is to simplify this equation and get it into a form we can easily work with. We'll combine like terms and then start thinking about how to isolate x. So, with the substitution complete, we're well on our way to finding the solution. Let's keep the momentum going!
Simplifying the Equation
Now that we've performed the substitution, our equation looks like this:
x² + (x)² = 32
The next step is to simplify this equation. First, let's get rid of those parentheses. (x)² is the same as x², so we can rewrite the equation as:
x² + x² = 32
Now we have two x² terms. We can combine these like terms, just like we would with any algebraic expression. Think of it as having one x² and adding another x² to it. That gives us two x²s, so our equation becomes:
2x² = 32
See how much simpler it's getting? We've gone from an equation with two variables to a simple quadratic equation in one variable. The next step is to isolate x². To do that, we'll divide both sides of the equation by 2. This will get rid of the coefficient in front of the x² term, leaving us with x² on its own. So, let's divide both sides by 2 and see what we get.
Solving for x
Okay, we're on the home stretch for finding the value(s) of x. We left off with the simplified equation:
2x² = 32
To isolate x², we need to divide both sides of the equation by 2. When we do that, we get:
x² = 16
Now we're getting somewhere! We have x² equal to a number. To find x, we need to take the square root of both sides of the equation. Remember, when we take the square root of a number, we get two possible solutions: a positive one and a negative one. This is super important because it means we're likely to find two solutions for x, which will lead to two solutions for our system of equations.
So, let's take the square root of both sides:
√(x²) = ±√16
This simplifies to:
x = ±4
This means we have two possible values for x: x = 4 and x = -4. We already know about the solution when x = -4 (which is (-4, -4)), so now we need to focus on the case when x = 4. This is where we'll use our other equation to find the corresponding y value. So, let's move on to the next step and find out what y is when x = 4.
Finding the Corresponding y Value
We've nailed down the two possible values for x: x = 4 and x = -4. We already know the solution corresponding to x = -4 is (-4, -4). Now, let's find the y value that goes with x = 4. This is where our second equation comes in super handy. Remember, it's:
y = x
This equation couldn't be simpler! It tells us that y is equal to x. So, if x = 4, then:
y = 4
There you have it! When x = 4, y = 4. This gives us our second solution to the system of equations. We now have both the x and y values that satisfy both equations. All that's left is to write out our solution and celebrate our success! So, let's put it all together and state our final answer.
Stating the Solution
We've done it! We've successfully used substitution to solve the system of equations.
Our system was:
x² + y² = 32y = x
We found two solutions:
- When
x = -4,y = -4, giving us the solution(-4, -4). - When
x = 4,y = 4, giving us the solution(4, 4).
The problem told us that one solution is (-4, -4), and it asked us to find the other solution. We've done just that! The other solution to the system of equations is (4, 4). So, we can confidently fill in those blanks. Guys, you've walked through the entire process of solving a system of equations using substitution. You've seen how to substitute, simplify, solve for one variable, and then find the other. This is a powerful technique that you can use in many different situations. Keep practicing, and you'll become a pro at solving systems of equations!
So, we've journeyed through the world of substitution and conquered a system of equations! Remember, substitution is a fantastic tool for solving systems, especially when one equation is already solved for a variable or can be easily rearranged. We took a system with a circle equation and a line, used substitution to simplify it, and found both solutions. This process not only helps us find the answers but also deepens our understanding of how equations and variables interact.
The key takeaways here are: look for opportunities to substitute, simplify carefully, and remember to find all possible solutions. Math can sometimes seem like a puzzle, but with the right tools and a methodical approach, you can solve it. So, keep practicing, keep exploring, and keep that mathematical curiosity alive! You've got this! Remember, every problem you solve makes you a stronger mathematician. And who knows? Maybe next time, you'll be the one explaining substitution to someone else!
