Solving A System Of Equations: A Step-by-Step Guide

by Alex Johnson 52 views

In this article, we'll walk through the process of solving a system of equations. We'll specifically tackle the system:

  • x = 2y
  • x² + y² - 2xy - 9 = 0

Solving systems of equations is a fundamental skill in algebra and calculus, with applications in various fields like physics, engineering, and economics. So, let's dive in and break down how to solve this particular system!

Understanding the Equations

Before we jump into solving, let's take a closer look at our equations. The first equation, x = 2y, is a simple linear equation. It tells us that the value of x is always twice the value of y. This is a crucial piece of information that we'll use to our advantage.

The second equation, x² + y² - 2xy - 9 = 0, might look a bit more intimidating at first glance. However, with a little algebraic manipulation, we can simplify it. Notice that the first three terms, x² + y² - 2xy, resemble a perfect square trinomial. Recognizing patterns like these is key to efficient problem-solving in mathematics.

Step 1: Substitution

The most effective way to solve this system is by using the substitution method. Since we already know that x = 2y, we can substitute this expression for x in the second equation. This will leave us with a single equation in terms of y, which we can then solve.

Substituting x = 2y into x² + y² - 2xy - 9 = 0, we get:

(2y)² + y² - 2(2y)y - 9 = 0

Now, let's simplify this equation.

Step 2: Simplifying the Equation

Expanding the terms, we have:

4y² + y² - 4y² - 9 = 0

Combining like terms, we get:

y² - 9 = 0

This simplified equation is much easier to work with. It's a simple quadratic equation in the form of a difference of squares.

Step 3: Solving for y

To solve y² - 9 = 0, we can use several methods. One way is to add 9 to both sides, giving us:

y² = 9

Taking the square root of both sides, we get:

y = ±3

This means we have two possible values for y: y = 3 and y = -3. These are the y-coordinates of the solutions to our system of equations.

Step 4: Solving for x

Now that we have the values of y, we can find the corresponding values of x using the equation x = 2y.

For y = 3, we have:

x = 2(3) = 6

So, one solution is the point (6, 3).

For y = -3, we have:

x = 2(-3) = -6

So, the other solution is the point (-6, -3).

Step 5: Verifying the Solutions

It's always a good idea to verify our solutions by plugging them back into the original equations. This helps to ensure that we haven't made any mistakes along the way.

Let's check the solution (6, 3):

  • x = 2y => 6 = 2(3) => 6 = 6 (True)
  • x² + y² - 2xy - 9 = 0 => 6² + 3² - 2(6)(3) - 9 = 0 => 36 + 9 - 36 - 9 = 0 => 0 = 0 (True)

So, (6, 3) is indeed a solution.

Now let's check the solution (-6, -3):

  • x = 2y => -6 = 2(-3) => -6 = -6 (True)
  • x² + y² - 2xy - 9 = 0 => (-6)² + (-3)² - 2(-6)(-3) - 9 = 0 => 36 + 9 - 36 - 9 = 0 => 0 = 0 (True)

So, (-6, -3) is also a solution.

The Solutions

Therefore, the solutions to the system of equations are:

  • (6, 3)
  • (-6, -3)

These are the points where the two equations intersect on a graph. The first equation, x = 2y, represents a straight line, and the second equation, x² + y² - 2xy - 9 = 0, represents a hyperbola. The solutions we found are the points where this line and hyperbola intersect.

Alternative Approach: Recognizing the Perfect Square

As we mentioned earlier, the second equation contains a perfect square trinomial. We can rewrite the equation x² + y² - 2xy - 9 = 0 as:

(x - y)² - 9 = 0

This form makes it even clearer that we can solve for (x - y). Adding 9 to both sides, we get:

(x - y)² = 9

Taking the square root of both sides, we have:

x - y = ±3

Now we have two new equations:

  • x - y = 3
  • x - y = -3

We can solve each of these equations along with x = 2y as a system of equations. This alternative approach leads to the same solutions we found earlier: (6, 3) and (-6, -3).

Key Takeaways

  • Substitution: 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.
  • Simplifying Equations: Recognizing patterns and simplifying equations can make the problem much easier to solve. In this case, recognizing the perfect square trinomial was helpful.
  • Verification: Always verify your solutions by plugging them back into the original equations. This helps to catch any errors.
  • Multiple Approaches: Often, there are multiple ways to solve a math problem. Exploring different approaches can deepen your understanding and problem-solving skills.

Conclusion

Solving systems of equations is a fundamental skill in mathematics. By understanding the underlying concepts and practicing different techniques, you can tackle a wide range of problems. Remember to break down complex problems into smaller, manageable steps, and don't be afraid to explore different approaches. With practice, you'll become more confident and proficient in solving systems of equations.

For further learning and practice on solving systems of equations, you can explore resources like Khan Academy, which offers comprehensive lessons and exercises on this topic.