Solve Equations By Substitution: X + Y = 5, X - Y = 1

by Felix Dubois 54 views

Hey guys! Today, we're diving deep into the world of solving systems of equations using the substitution method. This is a fundamental concept in algebra, and mastering it will unlock a whole new level of mathematical problem-solving skills. We'll break down the method step-by-step, using the example system:

  • x + y = 5
  • x - y = 1

So, buckle up and let's get started!

Understanding Systems of Equations

Before we jump into the substitution method, let's quickly recap what a system of equations actually is. In essence, a system of equations is a set of two or more equations containing the same variables. The goal is to find values for these variables that satisfy all equations simultaneously. Think of it like a puzzle where you need to find the perfect combination of numbers that fit all the given conditions.

In our example, we have two equations with two variables, x and y. The solution to this system will be a pair of values (one for x and one for y) that make both equations true. Graphically, each equation represents a line, and the solution is the point where these lines intersect. The substitution method is one way to find this intersection point algebraically, offering a neat alternative to graphing or other methods like elimination.

Now, why is understanding systems of equations so important? Well, they pop up everywhere in real-world applications! From calculating the optimal mix of ingredients in a recipe to modeling the trajectory of a rocket, systems of equations are essential tools for representing and solving complex problems. So, by mastering this concept, you're not just learning math; you're equipping yourself with skills that can be applied across various fields.

The key takeaway here is that a system of equations represents a set of interconnected relationships, and solving it means finding the values that make all those relationships hold true. Whether you're dealing with simple linear equations or more complex scenarios, the underlying principle remains the same: finding the solution that satisfies the entire system. Let's move on to exploring how the substitution method helps us achieve this goal, making sure to break down each step in a way that's easy to grasp and apply. We'll start with the first crucial step: isolating one variable in one of the equations. This is the foundation upon which the entire substitution process rests, so let's make sure we understand it thoroughly before moving on to the next steps.

The Substitution Method: A Step-by-Step Approach

The substitution method is all about strategic manipulation. The core idea is to isolate one variable in one equation and then substitute its equivalent expression into the other equation. This eliminates one variable, leaving us with a single equation that we can easily solve. Let's break down the steps using our example:

Step 1: Isolate a Variable

The first step is to choose one equation and isolate one of the variables. This means getting one variable by itself on one side of the equation. Looking at our system:

  • x + y = 5
  • x - y = 1

It seems easier to isolate x in the second equation (x - y = 1). To do this, we simply add y to both sides:

x - y + y = 1 + y x = 1 + y

Now we have x expressed in terms of y. Remember, the goal here is to find the simplest way to express one variable using the other. In some cases, you might need to divide to fully isolate the variable, but in this example, adding y was all it took. Choosing the right variable to isolate can make the whole process smoother, so it's worth taking a moment to assess the equations and identify the easiest route.

This step is crucial because it sets the stage for the substitution itself. By isolating x, we've created an expression (1 + y) that represents x's value in terms of y. This expression is what we'll use in the next step to eliminate x from the other equation. The beauty of the substitution method lies in this transformation: turning a system of two equations into a single equation that we can solve directly. Now that we've successfully isolated x, we're ready to move on to the next step: substituting this expression into the other equation. This is where the magic really happens, as we'll see how eliminating one variable allows us to solve for the remaining one.

Step 2: Substitute

Now that we have x = 1 + y, we can substitute this expression for x in the other equation (the one we didn't use in step 1). This is crucial: we substitute into the equation x + y = 5.

Replacing x with (1 + y) gives us:

(1 + y) + y = 5

See what happened? We've eliminated x entirely and now have an equation with only y. This is the power of substitution! We've transformed a two-variable problem into a single-variable problem, which is much easier to solve.

The substitution step is the heart of the method, where we leverage the expression we derived in step 1 to simplify the system. By replacing one variable with its equivalent expression, we create a new equation that contains only one unknown. This equation is the key to unlocking the solution, as it allows us to directly solve for one of the variables. It's like peeling away a layer of complexity, revealing the underlying simplicity of the problem.

The careful selection of which variable to isolate in step 1 directly impacts the ease of this substitution. If we had chosen poorly, the resulting equation might be more complicated. But by strategically isolating x in the equation x - y = 1, we've created a straightforward substitution that leads to a clean equation in terms of y. Now that we have this single-variable equation, we're ready to solve for y in the next step. This is where the arithmetic comes into play, and we'll see how a few simple operations can lead us to the value of one of our variables.

Step 3: Solve for the Remaining Variable

Now we have the equation (1 + y) + y = 5. Let's solve for y:

1 + y + y = 5 1 + 2y = 5 2y = 4 y = 2

Great! We've found the value of y: y = 2. This step is where our algebraic skills come into play. We use the standard techniques for solving equations: combining like terms, isolating the variable, and performing inverse operations. The goal is to manipulate the equation until we have the variable by itself on one side, revealing its value.

The process of solving for the remaining variable is a direct consequence of the substitution we performed in the previous step. By eliminating one variable, we created an equation that can be solved using familiar algebraic methods. This is a testament to the power of the substitution method: it transforms a complex problem into a series of simpler, manageable steps.

It's worth noting that the equation we solve in this step might not always be a simple linear equation. Depending on the original system, we might encounter quadratic equations or other types of equations. However, the underlying principle remains the same: we use algebraic techniques to isolate the remaining variable and determine its value. Now that we've successfully solved for y, we're one step closer to the complete solution. All that's left is to find the value of x, which we can do by plugging our value for y back into one of the original equations. This is the final piece of the puzzle, and it brings us full circle, demonstrating the interconnectedness of the system of equations.

Step 4: Substitute Back to Find the Other Variable

We know y = 2. Now we substitute this value back into either of the original equations or the equation we derived in step 1 (x = 1 + y). Let's use x = 1 + y:

x = 1 + 2 x = 3

So, we've found that x = 3.

This step is often called