Solving Equations: A Step-by-Step Guide

Hey guys! Today, we're diving deep into the fascinating world of equation-solving, specifically tackling a system of equations that might seem a bit intimidating at first glance. But don't worry, we'll break it down step by step, making it super easy to understand. Our mission? To solve the system of equations:

• 2x + y = 3
• 4x² - y² + 2x + 3y = 16

This looks like a mix of linear and quadratic equations, which means we'll need to use a combination of techniques to find our solutions. Let’s get started!

Understanding the Equations

Before we jump into solving, let's take a moment to understand what we're dealing with. The first equation, 2x + y = 3, is a linear equation. This means that if we were to graph it, we'd get a straight line. Linear equations are generally easier to work with because they have a consistent rate of change.

The second equation, 4x² - y² + 2x + 3y = 16, is a bit more complex. Notice the x² and y² terms? These indicate that this is a quadratic equation, and it could represent a variety of curves, like a hyperbola or an ellipse. The presence of both x² and y² terms, along with the other terms, suggests that this isn't a simple quadratic equation in one variable, making our approach a bit more intricate.

When we have a system of equations like this, we're looking for the values of x and y that satisfy both equations simultaneously. Geometrically, these solutions represent the points where the graphs of the two equations intersect. Solving this system requires a strategic approach, and one of the most effective methods is substitution.

The Substitution Method: Our Key Strategy

The substitution method is a powerful technique for solving systems of equations, especially when one equation can be easily solved for one variable in terms of the other. In our case, the first equation, 2x + y = 3, is perfect for this. We can easily isolate y:

y = 3 - 2x

Now, we have an expression for y in terms of x. This is our golden ticket! We can substitute this expression into the second equation, effectively eliminating y from the equation and leaving us with an equation in just one variable, x. This is a crucial step because solving a single-variable equation is much simpler than dealing with two variables at once.

Think of it like this: we're taking the information we know about the relationship between x and y from the first equation and using it to simplify the second equation. By substituting, we transform a complex two-variable problem into a more manageable single-variable problem. This is a common strategy in algebra and is super useful in many different contexts.

Performing the Substitution

Alright, let's get our hands dirty and actually perform the substitution. We'll take our expression for y, which is y = 3 - 2x, and plug it into the second equation:

4x² - y² + 2x + 3y = 16

becomes

4x² - (3 - 2x)² + 2x + 3(3 - 2x) = 16

See what we did there? We replaced every instance of y in the second equation with (3 - 2x). Now, we have an equation that only involves x. But before we can solve for x, we need to simplify this equation. This means expanding the squared term and distributing the constants. This is where careful algebra comes into play, and it's super important to take your time and avoid making mistakes. One wrong sign or exponent can throw off the entire solution.

Simplifying the Equation

Now comes the fun part – simplifying! Let’s carefully expand and simplify the equation:

4x² - (3 - 2x)² + 2x + 3(3 - 2x) = 16

First, we need to expand (3 - 2x)². Remember that (a - b)² = a² - 2ab + b², so:

(3 - 2x)² = 3² - 2(3)(2x) + (2x)² = 9 - 12x + 4x²

Now, let's substitute this back into our equation:

4x² - (9 - 12x + 4x²) + 2x + 3(3 - 2x) = 16

Next, we distribute the negative sign and the 3:

4x² - 9 + 12x - 4x² + 2x + 9 - 6x = 16

Notice how the 4x² terms cancel each other out? This is great news because it simplifies our equation even further! Now, let's combine like terms:

(4x² - 4x²) + (12x + 2x - 6x) + (-9 + 9) = 16

0 + 8x + 0 = 16

So, we're left with:

8x = 16

Woohoo! We've simplified the equation down to a simple linear equation in x. This is a huge step forward.

Solving for x

Solving for x is now a piece of cake! We have the equation:

8x = 16

To isolate x, we simply divide both sides of the equation by 8:

x = 16 / 8

x = 2

Fantastic! We've found our first solution: x = 2. But remember, we're solving a system of equations, so we need to find the corresponding value of y as well. This is where our earlier expression for y in terms of x comes in handy.

Finding the Value of y

We know that x = 2, and we have the equation:

y = 3 - 2x

Now, we just substitute our value of x into this equation:

y = 3 - 2(2)

y = 3 - 4

y = -1

Awesome! We've found that y = -1. So, one solution to our system of equations is x = 2 and y = -1. This means that the point (2, -1) is an intersection point of the two curves represented by our equations. But hold on, we're not quite done yet. Since we started with a quadratic equation, there might be another solution. We need to check if there are any other values of x and y that satisfy both equations.

Checking for Other Solutions

In this particular case, because the equation simplified so nicely after the substitution, we only ended up with one linear equation to solve for x. This indicates that there's likely only one solution to the system. However, it's always a good practice to check our solution to make sure it works in both original equations. This helps us catch any potential errors we might have made along the way.

Let's plug x = 2 and y = -1 into our original equations:

• Equation 1: 2x + y = 3 2(2) + (-1) = 4 - 1 = 3 (This checks out!)
• Equation 2: 4x² - y² + 2x + 3y = 16 4(2)² - (-1)² + 2(2) + 3(-1) = 4(4) - 1 + 4 - 3 = 16 - 1 + 4 - 3 = 16 (This also checks out!)

Since our solution satisfies both equations, we can confidently say that it is a valid solution.

Final Solution

After all our hard work, we've arrived at the solution! The system of equations:

• 2x + y = 3
• 4x² - y² + 2x + 3y = 16

has one solution:

(x, y) = (2, -1)

This means that the point (2, -1) is the only point where the line represented by the first equation and the curve represented by the second equation intersect. We successfully used the substitution method to navigate through this problem, simplifying it step by step until we found our answer. Remember, the key to solving complex equations is to break them down into smaller, more manageable parts.

Key Takeaways

Let's recap the main points we've learned in this equation-solving adventure:

1. Substitution is your friend: The substitution method is a powerful tool for solving systems of equations, especially when you can easily isolate one variable in terms of another.
2. Simplify, simplify, simplify: After substituting, take the time to carefully simplify the equation. Expanding terms, combining like terms, and watching out for those pesky negative signs can make a huge difference.
3. Solve for one variable at a time: By substituting, we reduced our two-variable problem into a single-variable problem, making it much easier to solve.
4. Don't forget the other variable: Once you've found the value of one variable, remember to substitute it back into one of the original equations (or the expression you derived) to find the value of the other variable.
5. Check your solutions: Always, always, always check your solutions in the original equations to make sure they work. This helps you catch any mistakes and ensures your answer is correct.
6. Practice makes perfect: The more you practice solving equations, the better you'll become at recognizing patterns and choosing the right techniques.

Practice Problems

Want to put your new skills to the test? Try solving these systems of equations using the substitution method:

1. y = x + 1 x² + y² = 13
2. x - y = 3 x² + y = 9
3. 2x + y = 5 x² - y = 0

Solving equations can be challenging, but it's also super rewarding. Keep practicing, and you'll become an equation-solving pro in no time! You got this!