Solving Systems Of Equations: Find Ordered Pair Solutions

Hey math enthusiasts! Today, we're diving deep into the super cool world of solving systems of equations. Specifically, we're going to tackle a problem that involves finding the ordered pair solutions for a given system. You know, those $(x, y)$ values that make both equations in the system sing the same tune? It's like finding the secret handshake that unlocks the truth for both! We've got a juicy one here: $

$$\left\{\begin{array}{c} y = -x^2 + 34 \\ y = 3x - 20 \end{array}\right.$$

This bad boy combines a quadratic equation (that's the $y = -x^2 + 34$ part, with that sweet $x^2$ term making a parabola) and a linear equation (the $y = 3x - 20$ part, a straight line). Our mission, should we choose to accept it (and we totally should!), is to find where these two graphs intersect. The points of intersection are precisely the ordered pair solutions we're hunting for. This isn't just about numbers, guys; it's about visualizing the relationships between different mathematical functions and pinpointing the exact moments they agree. So, buckle up, grab your calculators (or your trusty pencils!), and let's get this math party started!

Understanding the Problem: When Parabolas Meet Lines

Alright, let's really get our heads around what we're dealing with here. We have a system of two equations, and the goal is to find the specific ordered pair solutions. What does that actually mean? It means we're looking for coordinate pairs $(x, y)$ that satisfy both equations simultaneously. Think of it like this: one equation represents a graceful parabola opening downwards (because of the negative $x^2$ term), and the other represents a straightforward line with a positive slope. These two shapes are going to cross paths, and we need to find the exact coordinates of those crossing points. The equation $y = -x^2 + 34$ describes all the points that lie on the parabola, and $y = 3x - 20$ describes all the points on the line. When we solve the system, we're finding the $(x, y)$ pairs that are present on both the parabola and the line at the same time. This is incredibly useful in real-world scenarios, like figuring out when a projectile (following a parabolic path) will reach a certain height (represented by a line). So, understanding the nature of these equations is the first step. The parabola $y = -x^2 + 34$ has its vertex at $(0, 34)$ and opens downwards. The line $y = 3x - 20$ has a y-intercept of $-20$ and a slope of $3$. These visual clues help us anticipate roughly how many solutions we might find (could be zero, one, or two intersection points). The core concept is substitution or elimination, methods we'll explore next to get those precise $(x, y)$ values.

The Substitution Strategy: A Powerful Approach

One of the most elegant ways to find ordered pair solutions for a system of equations like this is the substitution method. It's all about replacing one variable in one equation with an expression from the other equation. Since both of our equations are already solved for $y$ (meaning we have $y$ isolated on one side), this method becomes incredibly straightforward. We can simply set the two expressions for $y$ equal to each other. Why does this work? Because if $y$ is equal to $-x^2 + 34$, and $y$ is also equal to $3x - 20$, then it logically follows that $-x^2 + 34$ must be equal to $3x - 20$. This is the foundation of the substitution method. We're essentially saying, "Hey, whatever $y$ is in the first equation, it's also that same value in the second equation, so let's equate the right-hand sides." This process cleverly eliminates one of the variables ($y$ in this case) and leaves us with a single equation in terms of the other variable ($x$). This new equation will be a quadratic equation, which we know how to solve! So, let's put it into action. We take our two equations: $y = -x^2 + 34$ and $y = 3x - 20$. Because both are equal to $y$, we can write: $ -x^2 + 34 = 3x - 20 $ This single equation now holds the key to finding the $x$-values of our solutions. Once we solve this equation for $x$, we'll have the first part of our ordered pairs. Remember, the goal is to find $(x, y)$ pairs, so finding $x$ is just the first, albeit crucial, step. The beauty of substitution is that it simplifies a system of two variables into a single variable problem, making it much more manageable. It's like untangling a knot by focusing on one strand at a time. We're setting ourselves up to solve for $x$, and once we have that, getting $y$ will be a piece of cake!

Solving for x: The Heart of the Matter

Now that we've used the substitution method to set up our equation, it's time to roll up our sleeves and solve for $x$. Our equation is: $ -x^2 + 34 = 3x - 20 $ To solve a quadratic equation, we generally want to get all the terms on one side, setting the equation equal to zero. This is a standard procedure that makes it easier to apply factoring, the quadratic formula, or completing the square. Let's move all the terms to the right side to make the $x^2$ term positive, which is often more convenient. We add $x^2$ to both sides: $ 34 = x^2 + 3x - 20 $ Then, we subtract $34$ from both sides: $ 0 = x^2 + 3x - 20 - 34 $ Which simplifies to: $ 0 = x^2 + 3x - 54 $ Fantastic! We now have a standard quadratic equation in the form $ax^2 + bx + c = 0$, where $a=1$, $b=3$, and $c=-54$. Our next step is to find the values of $x$ that satisfy this equation. We have a few options here: factoring, using the quadratic formula, or completing the square. Factoring is usually the quickest if it's possible. We're looking for two numbers that multiply to $-54$ and add up to $3$. Let's brainstorm pairs of factors for $54$: $(1, 54)$, $(2, 27)$, $(3, 18)$, $(6, 9)$. Since the product is negative, one factor must be positive and the other negative. We want their sum to be positive $3$, so the larger factor should be positive. Let's test our pairs: $-1 + 54 = 53$, $-2 + 27 = 25$, $-3 + 18 = 15$, $-6 + 9 = 3$. Bingo! The numbers are $-6$ and $9$. So, we can factor our quadratic equation as: $ (x - 6)(x + 9) = 0 $ For this product to be zero, at least one of the factors must be zero. So, we set each factor equal to zero and solve for $x$: $ x - 6 = 0 ightarrow x = 6 $ $ x + 9 = 0 ightarrow x = -9 $ Wowza! We've found two possible $x$-values for our ordered pair solutions: $x = 6$ and $x = -9$. This means our system likely has two intersection points, which makes sense given we're dealing with a parabola and a line. These are the $x$-coordinates of our solutions!

Finding the Corresponding y-Values: Completing the Pair

We've successfully navigated the exciting territory of solving for $x$, and we've emerged with two potential $x$-values: $6$ and $-9$. But remember, guys, we're on a quest for ordered pair solutions, which means we need both an $x$ and a $y$ value for each solution. Luckily, finding the corresponding $y$-values is the relatively easy part now! We can substitute each $x$-value back into either of the original equations to find the matching $y$. The key here is that the $y$-value should be the same regardless of which original equation you choose, because these $x$-values are the solutions where both equations are true. For simplicity, let's use the linear equation, $y = 3x - 20$, because it's generally easier to calculate with than the quadratic one. Let's tackle $x = 6$ first:

Substitute $x = 6$ into $y = 3x - 20$:

y = 3(6) - 20 $ $ y = 18 - 20 $ $ y = -2 $ So, for $x = 6$, we get $y = -2$. This gives us our first potential **ordered pair solution**: $(6, -2)$. Now, let's do the same for our other $x$-value, $x = -9$: Substitute $x = -9$ into $y = 3x - 20$: $ y = 3(-9) - 20 $ $ y = -27 - 20 $ $ y = -47 $ So, for $x = -9$, we get $y = -47$. This gives us our second potential **ordered pair solution**: $(-9, -47)$. There you have it! We've found the two $(x, y)$ pairs that satisfy both equations. We have $(6, -2)$ and $(-9, -47)$. These are the coordinates where the parabola $y = -x^2 + 34$ and the line $y = 3x - 20$ intersect. It's always a good idea to double-check these solutions by plugging them back into the *other* original equation (the quadratic one, $y = -x^2 + 34$) to ensure they work. Let's do that quickly: Check $(6, -2)$ in $y = -x^2 + 34$: $ -2 = -(6)^2 + 34 $ $ -2 = -36 + 34 $ $ -2 = -2 $ Perfect! It checks out. Check $(-9, -47)$ in $y = -x^2 + 34$: $ -47 = -(-9)^2 + 34 $ $ -47 = -(81) + 34 $ $ -47 = -81 + 34 $ $ -47 = -47 $ Awesome! This one checks out too. So, we are super confident in our **ordered pair solutions**. ## Verification: Ensuring Accuracy of Solutions We've done the heavy lifting, guys, and we've arrived at our potential **ordered pair solutions**: $(6, -2)$ and $(-9, -47)$. But in the world of mathematics, especially when you're solving systems of equations, verification isn't just a suggestion; it's a crucial step to ensure you haven't made any sneaky calculation errors along the way. Think of it as a final quality check before you declare victory. We already did a quick check by plugging our $x$-values into the *other* original equation, but a thorough verification involves plugging the *entire* ordered pair $(x, y)$ back into *both* original equations. This ensures that both the $x$ and $y$ values work together correctly in each equation. Let's go through it systematically. **For the first ordered pair solution: $(6, -2)$** * **Check in the first equation:** $y = -x^2 + 34$ Substitute $x=6$ and $y=-2$: $ -2 = -(6)^2 + 34

$ -2 = -36 + 34 $
$ -2 = -2 $ 
This equation holds true. Great!

• Check in the second equation: $y = 3x - 20$ Substitute $x=6$ and $y=-2$:

  $$-2 = 3(6) - 20$$

  $$-2 = 18 - 20$$

  $$-2 = -2$$

  This equation also holds true. Fantastic!

Since $(6, -2)$ satisfies both equations, it is indeed a valid ordered pair solution for the system.

For the second ordered pair solution: $(-9, -47)$

• Check in the first equation: $y = -x^2 + 34$ Substitute $x=-9$ and $y=-47$:

  $$-47 = -(-9)^2 + 34$$

  Remember that $(-9)^2$ is $81$, not $-81$. So, we have:

  $$-47 = -(81) + 34$$

  $$-47 = -81 + 34$$

  $$-47 = -47$$

  This equation is also satisfied. Excellent!

• Check in the second equation: $y = 3x - 20$ Substitute $x=-9$ and $y=-47$:

  $$-47 = 3(-9) - 20$$

  $$-47 = -27 - 20$$

  $$-47 = -47$$

  This equation holds true as well. Amazing!

Both $(-9, -47)$ and $(6, -2)$ have passed the rigorous verification process. This confirms that these are the correct ordered pair solutions for the given system of equations. It's this kind of meticulous checking that builds confidence in your mathematical results. So, when you're solving these problems, don't skip the verification step – it's your best friend for accuracy!

Conclusion: The Ordered Pairs That Make Both Equations True

And there you have it, folks! We've successfully navigated the process of finding the ordered pair solutions for the system of equations: $

$$\left\{\begin{array}{c} y = -x^2 + 34 \\ y = 3x - 20 \end{array}\right.$$

By employing the powerful substitution method, we were able to equate the expressions for $y$, which led us to a quadratic equation in $x$. Solving this quadratic equation, $x^2 + 3x - 54 = 0$, yielded two distinct $x$-values: $x = 6$ and $x = -9$. These $x$-values represent the horizontal positions where the parabola and the line intersect. The next critical step was to find the corresponding $y$-values for each $x$. By substituting these $x$-values back into the simpler linear equation ($y = 3x - 20$), we found the respective $y$-coordinates. For $x=6$, we got $y=-2$, giving us the ordered pair $(6, -2)$. For $x=-9$, we found $y=-47$, resulting in the ordered pair $(-9, -47)$.

We didn't stop there, though! A true mathematician knows the importance of verification. We meticulously plugged both $(6, -2)$ and $(-9, -47)$ back into both original equations. Each pair satisfied both the quadratic and the linear equation, confirming that these are indeed the correct ordered pair solutions. These solutions represent the points of intersection between the graph of the parabola $y = -x^2 + 34$ and the line $y = 3x - 20$. Finding these points is a fundamental skill in algebra and has wide-ranging applications in science, engineering, and economics where systems of equations model real-world phenomena. So, remember these steps: set up the equation by substitution, solve for one variable, find the corresponding values of the other variable, and always, always verify your answers. Keep practicing, and you'll become a pro at finding these intersection points!