Solving The System Of Equations Y=-(x+2)^2+1 And Y=4x+9
Hey guys! Today, we're diving deep into the world of system of equations, specifically tackling the challenge of finding the solutions for the system:
and
It might look intimidating at first, but don't worry, we'll break it down step-by-step, making sure everyone understands the underlying concepts and the solution process. We'll not only identify the correct answer but also explore why the other options are incorrect, giving you a solid grasp of how to approach similar problems in the future. So, grab your pencils, and let's get started!
Understanding the Equations
Before we jump into solving, let's take a closer look at the equations we're dealing with. The first equation, $y=-(x+2)^2+1$, represents a parabola. Parabolas are U-shaped curves, and this one opens downwards because of the negative sign in front of the squared term. The "+2" inside the parentheses shifts the parabola horizontally, and the "+1" at the end shifts it vertically. Understanding these transformations helps us visualize the graph and anticipate the solutions.
The second equation, $y=4x+9$, is a linear equation. This equation represents a straight line with a slope of 4 and a y-intercept of 9. The slope tells us how steep the line is, and the y-intercept is where the line crosses the y-axis. Visualizing both the parabola and the line is key to understanding how they might intersect, which will give us the solutions to the system.
In essence, solving a system of equations means finding the points where the graphs of the equations intersect. These points of intersection represent the (x, y) pairs that satisfy both equations simultaneously. There are several methods we can use to find these points, including substitution, elimination, and graphing. For this particular problem, we'll primarily use the substitution method, as it's a powerful and efficient way to solve systems where one variable is already isolated in one of the equations. By substituting one equation into the other, we can reduce the system to a single equation with a single variable, which is much easier to solve. Once we find the value(s) of that variable, we can then substitute back into either of the original equations to find the corresponding value(s) of the other variable.
Solving by Substitution
The substitution method is our main tool here. Since both equations are already solved for y, we can set them equal to each other. This gives us:
Now, let's simplify and solve for x. First, we need to expand the squared term:
Next, distribute the negative sign:
Combine like terms on the left side:
Now, let's move all the terms to one side to get a quadratic equation. Add $x^2$, $4x$, and 3 to both sides:
We now have a quadratic equation in the standard form. To solve it, we can either factor it, use the quadratic formula, or complete the square. Factoring is often the easiest method if the quadratic expression can be factored easily. In this case, we're in luck – the quadratic expression factors nicely. We're looking for two numbers that multiply to 12 and add up to 8. Those numbers are 6 and 2. So, we can factor the equation as follows:
To find the values of x that make the equation true, we set each factor equal to zero:
Solving these equations gives us:
These are the x-coordinates of the points where the parabola and the line intersect. To find the corresponding y-coordinates, we need to substitute these x-values back into either of the original equations. The linear equation, $y = 4x + 9$, is usually the easier choice for this step, as it involves less computation. Let's substitute each x-value into the linear equation.
For $x = -6$:
So, one solution is the point $(-6, -15)$.
For $x = -2$:
So, the other solution is the point $(-2, 1)$.
Therefore, the solutions to the system of equations are $(-6, -15)$ and $(-2, 1)$.
Analyzing the Answer Choices
Now that we've found the solutions, let's look at the answer choices provided:
- A. $(-2,1)$ and $(6,-15)$
- B. $(-2,1)$ and $(-6,-15)$
- C. $(2,1)$ and $(6,-15)$
- D. Discussion category
Comparing our solutions with the options, we can see that option B $(-2,1)$ and $(-6,-15)$ matches our findings. Thus, option B is the correct answer.
Let's also take a moment to discuss why the other options are incorrect. Option A has $(6, -15)$, which has a positive 6 for the x-coordinate. If we substitute x = 6 into the linear equation, we get y = 4(6) + 9 = 33, not -15. So, (6, -15) is not a solution. Option C has both incorrect x-coordinates. Substituting x = 2 into the linear equation gives y = 4(2) + 9 = 17, not 1. And as we've already discussed, (6, -15) is also not a solution. Option D is simply a category, not a solution.
By carefully solving the system and checking our answers against the options, we can confidently select the correct solution and understand why the others don't work.
Common Mistakes and How to Avoid Them
When solving systems of equations, especially those involving quadratic expressions, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and increase your accuracy.
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Incorrectly Expanding Squared Terms: One of the most frequent errors occurs when expanding expressions like $(x + 2)^2$. Remember that $(x + 2)^2$ is not equal to $x^2 + 4$. You need to use the FOIL method (First, Outer, Inner, Last) or the binomial theorem to expand it correctly: $(x + 2)^2 = (x + 2)(x + 2) = x^2 + 2x + 2x + 4 = x^2 + 4x + 4$. Failing to include the middle term (4x in this case) will lead to an incorrect quadratic equation and, consequently, incorrect solutions.
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Forgetting to Distribute the Negative Sign: In our problem, we had the expression $-(x + 2)^2$. After expanding the squared term, it's crucial to distribute the negative sign correctly. This means changing the sign of every term inside the parentheses. For example, $-(x^2 + 4x + 4)$ becomes $-x^2 - 4x - 4$. A common mistake is to only change the sign of the first term, which will throw off the entire solution process.
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Factoring Errors: When solving quadratic equations by factoring, it's essential to find the correct factors. Make sure the factors you choose multiply to the constant term and add up to the coefficient of the linear term. For instance, in our equation $x^2 + 8x + 12 = 0$, we needed factors that multiply to 12 and add up to 8. The correct factors are 6 and 2, but other pairs, like 3 and 4, might initially seem plausible. Always double-check your factors to ensure they satisfy both conditions.
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Only Finding x-values: A system of equations solution consists of both x and y values that satisfy both equations. Once you've solved for x, don't forget to substitute those values back into one of the original equations to find the corresponding y-values. Leaving out the y-values means you haven't fully solved the system.
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Substituting into the Wrong Equation: When finding the y-values, you can substitute the x-values into either of the original equations. However, it's generally easier to use the simpler equation, which is often the linear equation. Substituting into the more complex equation (in our case, the quadratic equation) increases the chance of making a calculation error.
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Not Checking Solutions: After finding potential solutions, it's a good practice to check them by substituting both the x and y values into both original equations. This ensures that the solutions are valid and helps catch any calculation errors made along the way. If a solution doesn't satisfy both equations, it's not a valid solution.
By keeping these common mistakes in mind and taking the time to double-check your work, you can significantly improve your accuracy when solving systems of equations.
Alternative Methods for Solving
While we primarily used the substitution method in this case, it's worth noting that other methods can also be used to solve systems of equations. Understanding these alternative approaches can provide you with a broader toolkit for tackling different types of problems.
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Graphing: Graphing is a visual method for solving systems of equations. You graph both equations on the same coordinate plane, and the points where the graphs intersect are the solutions to the system. Graphing can be particularly useful for understanding the nature of the solutions (e.g., whether there are zero, one, or multiple solutions). However, it may not always provide precise solutions, especially if the intersection points have non-integer coordinates. In our example, you could graph the parabola $y = -(x + 2)^2 + 1$ and the line $y = 4x + 9$. The points where the parabola and line intersect would visually represent the solutions we found algebraically. You can use graphing calculators or online tools like Desmos to create accurate graphs.
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Elimination: The elimination method involves manipulating the equations in the system so that when you add or subtract them, one of the variables is eliminated. This leaves you with a single equation in one variable, which you can solve. The elimination method is particularly useful when the equations are in standard form (Ax + By = C). In our specific problem, the substitution method was more straightforward because both equations were already solved for y. However, if the equations were given in a different form, the elimination method might be a viable option. For example, if we rewrote the equations as $y + (x + 2)^2 = 1$ and $y - 4x = 9$, we could subtract the second equation from the first to eliminate y, but this would still require expanding and simplifying, making substitution the more efficient choice here.
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Using Technology: Various technological tools, such as calculators and computer algebra systems (CAS), can solve systems of equations. These tools can be especially helpful for complex systems or when you need to solve a large number of systems. However, it's crucial to understand the underlying mathematical concepts, even when using technology. Relying solely on technology without understanding the methods can lead to errors or an inability to solve problems in situations where technology is not available (e.g., during a test). Graphing calculators can quickly graph the equations and find the intersection points, and CAS software like Mathematica or Maple can solve systems symbolically, providing exact solutions.
By being familiar with these alternative methods, you can choose the most efficient approach for each problem and have backup strategies in case one method proves difficult.
Conclusion
So, guys, we've successfully navigated the world of systems of equations and found that the solutions to our given system are $(-2, 1)$ and $(-6, -15)$, making option B the correct answer. Remember, the key is to understand the equations, choose the right method, and avoid common mistakes. Keep practicing, and you'll become a pro at solving these problems! And always double check your work! It helps eliminate making small mistakes.
