Finding The Intersection Point Solving Linear Equations
Hey guys! Today, we're diving into a super important concept in mathematics: finding the point where two linear equations intersect. This is a skill that pops up everywhere, from basic algebra to more advanced topics, and even in real-world applications. So, let's break it down and make sure we've got a solid understanding. We'll use the system of equations you provided as our example:
5x + y = 8
x + 3y = 9
We're looking for the (x, y) coordinates where these two lines cross each other. There are a couple of ways we can tackle this: the substitution method and the elimination method. Let's explore both!
Understanding Linear Equations and Intersections
Before we jump into solving, let's quickly recap what linear equations are and what it means for them to intersect. A linear equation is basically an equation that, when graphed, forms a straight line. Think of it as a relationship between two variables (usually x and y) where the highest power of the variables is 1. Our equations, 5x + y = 8 and x + 3y = 9, perfectly fit this description. They're both linear!
When we graph these equations, we get two lines. The intersection point is the exact spot where these lines cross each other. This point is special because it's the only (x, y) coordinate pair that satisfies both equations simultaneously. In other words, if you plug the x and y values of the intersection point into both equations, they will both be true. That's our goal: to find those magic x and y values.
Why is this important? Well, intersections pop up in tons of real-world scenarios. Imagine you're comparing two different phone plans, each with a different monthly fee and per-minute charge. The intersection point would tell you the number of minutes where the total cost of both plans is the same. Pretty neat, right? Or think about supply and demand curves in economics; the intersection point represents the market equilibrium. So, mastering this skill is super useful!
Method 1: The Substitution Method
The substitution method is all about isolating one variable in one equation and then substituting that expression into the other equation. This might sound a bit complicated, but it's actually pretty straightforward once you get the hang of it. Let's walk through it step-by-step using our equations:
5x + y = 8
x + 3y = 9
Step 1: Choose an equation and isolate a variable.
Look for the easiest variable to isolate. In this case, the 'y' in the first equation (5x + y = 8) seems like a good candidate because it has a coefficient of 1. Let's isolate it:
y = 8 - 5x
We simply subtracted 5x from both sides of the equation. Now we have an expression for 'y' in terms of 'x'.
Step 2: Substitute the expression into the other equation.
Now, we take the expression we just found (y = 8 - 5x) and substitute it into the other equation (x + 3y = 9). It's crucial to use the other equation here; otherwise, we'll just end up with a tautology (something that's always true but doesn't help us solve anything).
x + 3(8 - 5x) = 9
See what we did? We replaced 'y' with the expression '8 - 5x'. Now we have a single equation with only one variable ('x'), which we can solve!
Step 3: Solve for the remaining variable.
Let's simplify and solve for 'x':
x + 24 - 15x = 9
-14x + 24 = 9
-14x = -15
x = 15/14
So, we've found the x-coordinate of our intersection point! It's 15/14. Don't be scared of fractions; they're just numbers too!
Step 4: Substitute the value back to find the other variable.
Now that we know x = 15/14, we can plug it back into either of the original equations (or even the expression y = 8 - 5x) to find 'y'. Let's use y = 8 - 5x because it's already set up for us:
y = 8 - 5(15/14)
y = 8 - 75/14
y = (112 - 75) / 14
y = 37/14
Alright! We've got our y-coordinate: 37/14.
Step 5: Write the solution as an ordered pair.
Finally, we write our solution as an ordered pair (x, y): (15/14, 37/14). This is the point where the two lines intersect!
The substitution method is a powerful tool, especially when one of the variables is easy to isolate. But what if isolating a variable is a bit messy? That's where the elimination method comes in!
Method 2: The Elimination Method
The elimination method is another fantastic way to solve systems of linear equations. It's particularly useful when neither variable is easily isolated. The idea here is to manipulate the equations so that the coefficients of one variable are opposites (e.g., 3 and -3). Then, when we add the equations together, that variable will be eliminated, leaving us with a single equation in one variable. Let's see how it works with our example:
5x + y = 8
x + 3y = 9
Step 1: Choose a variable to eliminate.
We can choose to eliminate either 'x' or 'y'. Let's go for 'y' this time. To eliminate 'y', we need to make the coefficients of 'y' in both equations opposites. Currently, they are 1 and 3. We can multiply the first equation by -3 to get a -3y term:
-3(5x + y) = -3(8)
-15x - 3y = -24
Now our system of equations looks like this:
-15x - 3y = -24
x + 3y = 9
Notice that the 'y' terms now have coefficients of -3 and 3, which are opposites!
Step 2: Add the equations together.
Now, we add the two equations together, term by term:
(-15x - 3y) + (x + 3y) = -24 + 9
-14x = -15
The 'y' terms canceled out, just as we planned! We're left with a simple equation in 'x'.
Step 3: Solve for the remaining variable.
Solving for 'x', we get:
x = -15 / -14
x = 15/14
Hey, that's the same x-coordinate we got using the substitution method! That's a good sign; it means we're on the right track.
Step 4: Substitute the value back to find the other variable.
Just like with the substitution method, we plug x = 15/14 back into either of the original equations to find 'y'. Let's use the second equation, x + 3y = 9:
(15/14) + 3y = 9
3y = 9 - 15/14
3y = (126 - 15) / 14
3y = 111/14
y = (111/14) / 3
y = 37/14
And there it is! We found y = 37/14, which matches our previous result.
Step 5: Write the solution as an ordered pair.
Our solution, as an ordered pair, is (15/14, 37/14). This confirms that both methods lead to the same intersection point.
Comparing the Methods: Substitution vs. Elimination
So, we've seen two powerful methods for finding the intersection point of linear equations. Which one should you use? Well, it depends on the specific equations you're dealing with.
- Substitution Method: This method is great when one of the variables has a coefficient of 1 (or -1) because it's easy to isolate that variable. If you see an equation like 'y = ...' or 'x = ...', substitution is often a good choice.
- Elimination Method: This method shines when the coefficients of one of the variables are already opposites or can be easily made opposites by multiplying one or both equations by a constant. If you see terms like '2x' and '-2x' or 'y' and '-y', elimination might be the faster route.
In reality, both methods will always work, so it's really about choosing the one that feels most efficient and comfortable for you. The more you practice, the better you'll become at recognizing which method is best suited for a given problem.
Putting It All Together
Let's recap the key steps for finding the intersection point of two linear equations:
- Choose a method: Decide whether substitution or elimination seems more appropriate for the given equations.
- Apply the method:
- Substitution: Isolate one variable in one equation, substitute the expression into the other equation, solve for the remaining variable, and then substitute back to find the other variable.
- Elimination: Multiply one or both equations by constants to make the coefficients of one variable opposites, add the equations together to eliminate that variable, solve for the remaining variable, and then substitute back to find the other variable.
- Write the solution: Express the solution as an ordered pair (x, y).
- Check your answer (optional but recommended): Plug the x and y values back into both original equations to make sure they hold true. This helps catch any errors you might have made along the way.
Finding the intersection point of two linear equations is a fundamental skill in algebra and beyond. By mastering both the substitution and elimination methods, you'll be well-equipped to tackle a wide range of problems. So, keep practicing, and you'll become a pro in no time! You got this, guys!
Therefore, the intersection point of the two linear equations is:
