Solving Systems Of Equations: Find The Ordered Pair Solution

by Alex Johnson 61 views

Hey there! Let's dive into the fascinating world of linear equations and learn how to find their solutions. Specifically, we're going to tackle the question of how to find the ordered pair that satisfies a system of two linear equations. It might sound a bit intimidating at first, but trust me, it's like solving a fun puzzle! In this article, we’ll break down the steps with an example. So, let’s get started and unlock the secrets of these equations together.

Understanding Systems of Linear Equations

Before we jump into solving, let's make sure we're all on the same page about what a system of linear equations actually is. A system of linear equations is simply a set of two or more linear equations that share the same variables. Think of it as a team of equations working together. Each equation represents a straight line on a graph, and the solution to the system is the point (or points) where these lines intersect. This point of intersection is the ordered pair (x, y) that makes all the equations in the system true.

To really grasp this, consider two lines crisscrossing on a graph. The spot where they meet? That's the solution we're after! It's the special coordinate that fits perfectly into both equations, making them balance just right. So, when we talk about solving systems of equations, we're essentially on a quest to pinpoint these crucial intersection points. Keep this visual in mind as we move forward – it'll make the process much clearer and more intuitive.

Now, why do we even care about solving these systems? Well, they pop up everywhere in the real world! From figuring out the break-even point for a business to predicting traffic flow, systems of equations are powerful tools. They help us model situations where multiple conditions need to be met simultaneously. Think of it like this: imagine you're trying to decide between two phone plans, each with different pricing structures. Systems of equations can help you determine when one plan becomes cheaper than the other, ensuring you make the smartest choice for your needs.

Methods for Solving Systems of Equations

There are several ways to solve systems of linear equations, but two of the most common methods are the substitution method and the elimination method. Each has its strengths, making some problems easier to tackle than others. Let's take a closer look at each one.

  • Substitution Method: The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This effectively turns the system of two equations with two variables into a single equation with one variable, which we can then solve. Once we find the value of one variable, we can plug it back into either of the original equations to find the value of the other variable.

    Think of it like this: imagine you have two puzzle pieces that fit together to form a picture. The substitution method is like taking one piece, tracing its outline, and then fitting that outline into the other piece. This creates a new, single puzzle that's much easier to solve! For example, if you have the equations y = 2x + 1 and 3x + y = 6, you could substitute the expression for y from the first equation into the second equation, giving you 3x + (2x + 1) = 6. Now you have one equation with just x, which you can easily solve.

  • Elimination Method: The elimination method, also known as the addition method, involves manipulating the equations so that the coefficients of one of the variables are opposites. When you add the equations together, that variable is eliminated, leaving you with a single equation in one variable. Again, you can solve for this variable and then substitute back into one of the original equations to find the other variable.

    The elimination method is like a strategic subtraction. Imagine you have two groups of objects, and you want to know the difference in the number of a specific object between the two groups. By carefully subtracting the groups, you can eliminate the other objects and focus solely on the one you're interested in. For instance, if you have the equations 2x + y = 7 and x - y = 2, notice that the y terms have opposite signs. Adding the equations together immediately eliminates y, giving you 3x = 9, which is simple to solve for x.

Choosing between substitution and elimination often comes down to personal preference and the structure of the equations themselves. If one equation is already solved for a variable, substitution might be the quicker route. If the coefficients of one variable are easily made opposites, elimination might be the way to go. With practice, you'll develop a sense for which method is best suited for a particular problem. But, just like having different tools in a toolbox, knowing both methods gives you the flexibility to tackle any system of equations that comes your way.

Step-by-Step Solution

Now, let's apply these methods to the system of equations we have:

  1. y = -7x + 2
  2. y = 9x - 14

Since both equations are already solved for y, the substitution method seems like a natural fit here. We can set the expressions for y equal to each other, which means we're essentially saying that the point where the lines intersect has the same y-value in both equations. This is a crucial step in finding that special ordered pair that solves the system.

  • Step 1: Set the equations equal to each other.

    Because both equations are solved for y, we can set the right-hand sides equal: -7x + 2 = 9x - 14. This is where the magic of substitution happens – we've transformed our two-equation problem into a single equation that we can solve for x.

  • Step 2: Solve for x.

    Now, let’s isolate x. Add 7x to both sides of the equation: 2 = 16x - 14. Then, add 14 to both sides: 16 = 16x. Finally, divide both sides by 16: x = 1. We've found our x-coordinate! This is a huge step forward – we're halfway to finding the ordered pair that solves the system.

  • Step 3: Substitute the value of x back into one of the original equations to solve for y.

    We can use either equation, but let's use the first one: y = -7x + 2. Substitute x = 1: y = -7(1) + 2. Simplify: y = -7 + 2 = -5. We've found our y-coordinate as well! It's like connecting the dots – we now have both the x and y values that make our equations balance perfectly.

  • Step 4: Write the solution as an ordered pair.

    The solution is (1, -5). This is the point where the two lines intersect on a graph. It's the single, unique solution that satisfies both equations simultaneously. Think of it as the key that unlocks the puzzle of the system.

  • Step 5: Check the solution by substituting it into both original equations.

    This is a crucial step to ensure we haven't made any mistakes along the way. Let's plug (1, -5) into both equations:

    • Equation 1: y = -7x + 2 becomes -5 = -7(1) + 2, which simplifies to -5 = -5. This checks out!
    • Equation 2: y = 9x - 14 becomes -5 = 9(1) - 14, which simplifies to -5 = -5. This also checks out!

    Since the ordered pair (1, -5) satisfies both equations, we can confidently say that it is indeed the solution to the system. This final check gives us the peace of mind that we've solved the problem correctly. Congratulations, you've successfully navigated the world of systems of linear equations!

Common Mistakes to Avoid

Solving systems of equations can be tricky, and it’s easy to make small errors that lead to the wrong answer. But don't worry! By being aware of these common pitfalls, you can avoid them and boost your problem-solving confidence. Let’s highlight a few key areas where mistakes often happen.

  • Incorrectly distributing a negative sign: This is a classic algebra mistake that can sneak into your work, especially when using the substitution or elimination method. For example, if you have an equation like -(x - 3), remember that the negative sign needs to be distributed to both terms inside the parentheses, resulting in -x + 3. Forgetting to distribute the negative sign correctly can throw off your entire solution, so always double-check those signs!

  • Combining like terms incorrectly: When simplifying equations, it's crucial to combine only like terms. You can't combine a term with x with a constant term, for instance. Make sure you're adding or subtracting the coefficients of the correct terms. A little extra attention here can save you from unnecessary errors.

  • Substituting back into the wrong equation: After solving for one variable, you need to substitute that value back into one of the original equations to find the other variable. A common mistake is to substitute back into an equation you've already manipulated, which can sometimes lead to incorrect results. Always go back to the original equations to ensure accuracy. It’s like double-checking your route on a map to make sure you’re still on the right path!

  • Not checking the solution: This is perhaps the most important mistake to avoid. Always, always, always check your solution by substituting the ordered pair into both original equations. This is your safety net! If the solution doesn't satisfy both equations, you know you've made a mistake somewhere and need to go back and review your steps. Think of it as proofreading your work before submitting it – it's a crucial step in the problem-solving process.

By keeping these common mistakes in mind and double-checking your work, you'll be well on your way to mastering systems of equations. Remember, practice makes perfect, so the more you solve, the more comfortable and confident you'll become!

Conclusion

So, there you have it! We've successfully navigated the world of solving systems of linear equations. We've explored the substitution method, step-by-step, and learned how to find the ordered pair that satisfies a system of equations. Remember, the key is to break down the problem into manageable steps, double-check your work, and don't be afraid to practice. You've got this!

Linear equations might seem like abstract concepts, but they're incredibly useful tools in many real-world situations. From calculating costs to predicting trends, the ability to solve systems of equations opens up a world of possibilities. The skills you've learned here will serve you well in future math courses and beyond. So keep practicing, keep exploring, and keep that problem-solving spirit alive!

For further exploration and practice, check out Khan Academy's Systems of Equations resources, a fantastic resource for learning and mastering mathematical concepts.