Find The Ordered Pair Solution For Linear Equations

Hey math whizzes and problem-solvers! Today, we're diving deep into the awesome world of linear equations and figuring out exactly which ordered pair solution makes a system of equations true. You know, those situations where you've got two or more equations with the same variables, and you're trying to find that sweet spot, that one specific point $(c, d)$ where both equations sing the same tune. It's like solving a puzzle, and the reward is finding that unique intersection. We're going to tackle a specific system and walk through how to find that elusive solution, so get ready to flex those algebra muscles!

Our challenge today is to solve the following system:

${egin{aligned}-c+2 d &=13 -9 c-4 d &=-15\end{aligned}$

We're given four potential ordered pairs: $(-1,6)$, $(-1,7)$, $(6,-1)$, and $(7,10)$. Your mission, should you choose to accept it, is to determine which of these pairs is the true solution to this system. Remember, a solution to a system of equations is an ordered pair that satisfies all equations in the system simultaneously. This means when you plug the values of $c$ and $d$ from the ordered pair into both equations, both equations must result in a true statement.

Let's break down the strategies for solving this. We've got a couple of rock-solid methods up our sleeves: substitution and elimination. Both are super effective, and the one you choose often comes down to personal preference or what looks easiest given the coefficients.

The Substitution Method: This is where you isolate one variable in one equation and then substitute that expression into the other equation. For our system, we could easily isolate '$c$' from the first equation: $-c = 13 - 2d$, which means $c = -13 + 2d$. Now, we'd take this expression for '$c$' and plug it into the second equation wherever we see '$c$'. This will give us an equation with only '$d$', which we can then solve. Once we have the value for '$d$', we can plug it back into our expression for '$c$' (or either of the original equations) to find the corresponding value of '$c$'. It’s a systematic way to peel back the layers of the problem.

The Elimination Method: This method is all about manipulating the equations (multiplying them by constants) so that when you add or subtract the equations, one of the variables cancels out. Looking at our system, the coefficients of '$d$' are $2$ and $-4$. If we multiply the first equation by $2$, the '$d$' term becomes $4d$, which is the exact opposite of the '-4d' in the second equation. Perfect for elimination! Multiplying the first equation by 2 gives us: $2(-c + 2d) = 2(13)$, resulting in $-2c + 4d = 26$. Now, we can add this modified first equation to the original second equation:

$(-2c + 4d) + (-9c - 4d) = 26 + (-15)$

See how the '$4d$' and '-4d' cancel each other out? This leaves us with $-11c = 11$. Solving for '$c$' is a piece of cake from here! Once we have '$c$', we can plug it back into either of the original equations to find '$d$'.

Checking the Potential Solutions: Another super efficient approach, especially when you're given multiple-choice options like we are, is to simply test each ordered pair. This method is often called direct substitution or plug-and-chug. You take each pair $(c, d)$ and substitute its values into both equations. If both equations hold true for a particular pair, then congratulations, you've found the solution! This can save a ton of time if one of the first few pairs you test happens to be the correct one. Let's try this with our first option, $(-1, 6)$.

Test Pair 1: $(-1, 6)$

• Equation 1: $-c + 2d = 13$ Substitute $c = -1$ and $d = 6$: $-(-1) + 2(6) = 1 + 12 = 13$. This equation holds true! (13 = 13)

• Equation 2: $-9c - 4d = -15$ Substitute $c = -1$ and $d = 6$: $-9(-1) - 4(6) = 9 - 24 = -15$. This equation also holds true! (-15 = -15)

Since the ordered pair $(-1, 6)$ satisfies both equations in the system, we've found our solution! It’s that simple, guys. No need to test the other options if you're confident in your substitutions. However, for the sake of thoroughness and learning, let's quickly see why the others don't work.

Test Pair 2: $(-1, 7)$

• Equation 1: $-c + 2d = 13$ Substitute $c = -1$ and $d = 7$: $-(-1) + 2(7) = 1 + 14 = 15$. This equation is false! (15 ≠ 13) Since it fails the first equation, we don't even need to check the second one. This pair is not the solution.

Test Pair 3: $(6, -1)$

• Equation 1: $-c + 2d = 13$ Substitute $c = 6$ and $d = -1$: $-(6) + 2(-1) = -6 - 2 = -8$. This equation is false! (-8 ≠ 13) Again, this pair is immediately eliminated.

Test Pair 4: $(7, 10)$

• Equation 1: $-c + 2d = 13$ Substitute $c = 7$ and $d = 10$: $-(7) + 2(10) = -7 + 20 = 13$. This equation holds true! (13 = 13)

• Equation 2: $-9c - 4d = -15$ Substitute $c = 7$ and $d = 10$: $-9(7) - 4(10) = -63 - 40 = -103$. This equation is false! (-103 ≠ -15) So, this pair is also not the solution.

As you can see, only the ordered pair $(-1, 6)$ works for both equations. This confirms our finding. It's awesome how these different methods all lead us to the same correct answer, right? Whether you prefer the direct substitution check, the elegance of elimination, or the step-by-step process of substitution, mastering these techniques is key to conquering systems of linear equations. Keep practicing, and you'll be solving them in your sleep!

Why This Matters in the Real World

Now, you might be thinking, "Okay, cool, I can solve for $c$ and $d$, but why does this even matter?" Great question! Systems of linear equations are actually everywhere. Think about resource allocation in a business, figuring out the best travel routes, or even in economics when trying to find market equilibrium. For instance, imagine a company produces two types of gadgets. Each gadget requires a certain amount of labor and raw materials. If you know the total labor hours and total raw materials available, and you know how much of each resource each gadget uses, you can set up a system of linear equations to find out exactly how many of each gadget to produce to use up all your resources. Or consider two different cell phone plans. Plan A charges a monthly fee plus a per-minute rate, and Plan B has a different structure. You could set up equations to find out at what number of minutes (or usage) the two plans cost the same amount. Finding the solution $(c, d)$ means finding that exact point of equivalence or optimal production. So, these aren't just abstract math problems; they're practical tools for decision-making in tons of different fields. Understanding how to solve them efficiently, like we just did by testing ordered pairs, is a super valuable skill that goes way beyond the classroom. It’s all about finding balance and understanding relationships between different quantities, which is a fundamental concept in math and life!

Deep Dive: The Geometry of Solutions

Let's get a little more visual, guys. Every linear equation in two variables, like $Ax + By = C$, represents a straight line on a coordinate plane. When we have a system of two linear equations, we are essentially looking at two lines on the same plane. The solution to the system, the ordered pair $(c, d)$, is the point where these two lines intersect. If the lines are parallel and never meet, the system has no solution. If the two equations actually represent the same line (they are dependent), then there are infinitely many solutions, and every point on that line is a solution. In our case, we found a single, unique solution, $(-1, 6)$. This means that if we were to graph the lines represented by $-c + 2d = 13$ and $-9c - 4d = -15$, they would cross each other at the exact point $(-1, 6)$.

To visualize this, let's find a couple of points for each line.

Line 1: $-c + 2d = 13$

• If $c= -1$, then $-(-1) + 2d = 13 ightarrow 1 + 2d = 13 ightarrow 2d = 12 ightarrow d = 6$. So, $(-1, 6)$ is on this line.
• If $d = 0$, then $-c + 2(0) = 13 ightarrow -c = 13 ightarrow c = -13$. So, $(-13, 0)$ is on this line.

Line 2: $-9c - 4d = -15$

• If $c = -1$, then $-9(-1) - 4d = -15 ightarrow 9 - 4d = -15 ightarrow -4d = -24 ightarrow d = 6$. So, $(-1, 6)$ is on this line.
• If $d = 0$, then $-9c - 4(0) = -15 ightarrow -9c = -15 ightarrow c = -15/-9 = 5/3$. So, $(5/3, 0)$ is on this line.

Plotting these points and drawing the lines would visually confirm that they intersect at $(-1, 6)$. This geometric interpretation really solidifies the concept of a system's solution as the common point shared by all its constituent equations. It’s a powerful way to understand algebraic relationships!

Conclusion: The Power of Precision

So there you have it, folks! We've explored how to find the ordered pair solution for a system of linear equations. Whether you used substitution, elimination, or the super-convenient method of testing the given options, we confirmed that $(-1, 6)$ is the unique solution to the system:

${egin{aligned}-c+2 d &=13 -9 c-4 d &=-15\end{aligned}$

This process highlights the importance of precision in mathematics. A small error in calculation can lead you to the wrong answer, so double-checking your work, especially when substituting values, is key. Remember, each method has its strengths, and choosing the right one can make solving problems much smoother. Keep practicing these skills, and you'll become a pro at navigating the world of linear equations and their solutions. Happy solving!