Solving System Of Equations: Y^2 = 8(x-1) & 2x - Y = 2

by Alex Johnson 55 views

Let's dive into how to solve this fascinating system of equations. We're presented with a combination of a parabola and a linear equation, which makes for an interesting mathematical journey. Our goal is to find the points where these two equations intersect, meaning the (x, y) values that satisfy both equations simultaneously. This article will guide you through a step-by-step process to tackle this problem, making sure you understand each move we make.

Understanding the Equations

Before we jump into solving, let's understand what we're working with. We have two equations:

  1. y² = 8(x - 1)
  2. 2x - y = 2

The first equation, y² = 8(x - 1), represents a horizontal parabola. You can recognize this because the y term is squared, and it's in the form of y² = 4a(x - h), where (h, 0) is the vertex of the parabola. In our case, the vertex is (1, 0), and the parabola opens to the right. The shape of a parabola is like a U, and understanding its orientation is crucial for visualizing the solutions. This equation tells us the relationship between x and y in a curved path.

The second equation, 2x - y = 2, is a linear equation. It represents a straight line. We can rewrite it in slope-intercept form (y = mx + b) to easily visualize it: y = 2x - 2. Here, the slope (m) is 2, and the y-intercept (b) is -2. Linear equations are straightforward; they depict a constant rate of change between x and y. This line intersects the y-axis at -2 and rises 2 units for every 1 unit increase in x.

Step-by-Step Solution

Now that we understand our equations, let's solve them. The most effective method here is substitution. Here’s how we'll do it:

1. Isolate a Variable in the Linear Equation

The linear equation, 2x - y = 2, is easier to manipulate. Let's isolate y. Add y to both sides and subtract 2 from both sides to get:

y = 2x - 2

2. Substitute into the Parabola Equation

Now, we'll substitute this expression for y into the parabola equation, y² = 8(x - 1). Replacing y with (2x - 2), we get:

(2x - 2)² = 8(x - 1)

This substitution transforms the equation into one with only x as the variable, which we can then solve.

3. Simplify and Solve for x

Let's expand and simplify the equation:

  • Expand (2x - 2)²: (2x - 2)² = (2x - 2)(2x - 2) = 4x² - 8x + 4
  • So, our equation becomes: 4x² - 8x + 4 = 8(x - 1)
  • Distribute the 8 on the right side: 4x² - 8x + 4 = 8x - 8
  • Move all terms to one side to set the equation to zero: 4x² - 8x + 4 - 8x + 8 = 0
  • Combine like terms: 4x² - 16x + 12 = 0

Now, we have a quadratic equation. We can simplify it further by dividing all terms by 4:

x² - 4x + 3 = 0

This quadratic equation is much easier to handle. Let's factor it:

(x - 3)(x - 1) = 0

Setting each factor equal to zero gives us the solutions for x:

  • x - 3 = 0 => x = 3
  • x - 1 = 0 => x = 1

So, we have two possible x-values: x = 3 and x = 1.

4. Find the Corresponding y-values

We have the x-values, but we need the corresponding y-values to complete our solutions. We'll use the linear equation y = 2x - 2 because it's simpler to work with:

  • For x = 3: y = 2(3) - 2 = 6 - 2 = 4 So, one solution is (3, 4).
  • For x = 1: y = 2(1) - 2 = 2 - 2 = 0 So, another solution is (1, 0).

5. Verify the Solutions

It's always a good idea to verify our solutions by plugging them back into both original equations:

  • For (3, 4):
    • Parabola: 4² = 8(3 - 1) => 16 = 8(2) => 16 = 16 (True)
    • Line: 2(3) - 4 = 2 => 6 - 4 = 2 => 2 = 2 (True)
  • For (1, 0):
    • Parabola: 0² = 8(1 - 1) => 0 = 8(0) => 0 = 0 (True)
    • Line: 2(1) - 0 = 2 => 2 - 0 = 2 => 2 = 2 (True)

Both solutions satisfy both equations, so they are correct.

The Solutions

We've successfully solved the system of equations! The solutions are:

  • (3, 4)
  • (1, 0)

These are the points where the parabola and the line intersect on a graph. Visually, if you were to graph these two equations, you would see the parabola and the line crossing at these two points.

Graphical Interpretation

Understanding the graphical representation of these solutions can solidify your understanding. Imagine plotting the parabola y² = 8(x - 1) and the line 2x - y = 2 on a coordinate plane. The points (1, 0) and (3, 4) are where these two graphs intersect.

The parabola opens to the right, and the line cuts through it at two distinct points. This graphical interpretation gives a visual confirmation of our algebraic solution. Graphing these equations can be a valuable exercise to enhance your understanding of how systems of equations work.

Alternative Methods

While substitution is a straightforward approach for this system, it's worth noting that other methods could be used, though they might be more complex in this particular case. For instance, you could try to solve for x in the linear equation and substitute that into the parabola equation. However, this would lead to a more complicated expression to solve.

Another potential method involves rearranging both equations and looking for common terms or factors that can help simplify the system. However, in this scenario, substitution provides a clear and efficient pathway to the solutions. Understanding alternative methods is beneficial, as different systems may be better suited to different techniques.

Common Mistakes to Avoid

When solving systems of equations, certain common mistakes can trip up even experienced mathematicians. Here are a few to watch out for:

  1. Incorrectly Expanding Squared Terms: A frequent error is mishandling the expansion of terms like (2x - 2)². Remember to use the correct formula or method to expand binomials to avoid algebraic errors.
  2. Forgetting to Distribute: When simplifying equations, always ensure you distribute terms correctly. For example, in the equation 4x² - 8x + 4 = 8(x - 1), make sure to distribute the 8 to both x and -1.
  3. Missing Solutions: When solving quadratic equations, make sure to consider both possible solutions. Factoring or using the quadratic formula carefully can help avoid missing a solution.
  4. Not Verifying Solutions: Always check your solutions by plugging them back into the original equations. This step is crucial to identify any errors made during the solving process.

By being mindful of these common pitfalls, you can increase your accuracy and confidence when tackling systems of equations.

Conclusion

Solving the system of equations y² = 8(x - 1) and 2x - y = 2 demonstrates a classic application of algebra. By using substitution and carefully simplifying the resulting equations, we found the solutions (3, 4) and (1, 0). Remember, the key to success in these problems lies in a clear understanding of algebraic principles and meticulous attention to detail. Practice is essential to mastering these skills, so keep tackling similar problems to build your confidence and proficiency.

From understanding the nature of parabolas and lines to executing the algebraic steps, we've covered the essential aspects of this problem. We hope this guide has been helpful in your mathematical journey! For further exploration of systems of equations, consider checking out resources like Khan Academy's Systems of Equations Section. Happy solving!