Finding Intersection: 2x - Y + 4 = 0 And X + Y - 7 = 0

Hey guys! Today, we're diving into a fun algebraic problem: finding the point where two lines intersect. Specifically, we're going to figure out the coordinates of the point that lies on both the line defined by the equation 2x - y + 4 = 0 and the line defined by x + y - 7 = 0. This is a classic problem in algebra, and mastering it will help you in various mathematical and real-world scenarios. So, grab your pencils, and let's get started!

Understanding the Problem

Before we jump into the solution, let's make sure we understand what the problem is asking. We have two linear equations, each representing a straight line on a coordinate plane. The solution we're looking for is the point (x, y) that satisfies both equations simultaneously. In other words, it's the point where the two lines cross each other. This point is crucial because it represents the common solution to both equations. Think of it as the 'sweet spot' that makes both statements true.

Why is this important?

Understanding how to find the intersection of lines isn't just an academic exercise. It has practical applications in various fields, including:

• Engineering: Determining structural stability or the optimal point for connections.
• Economics: Finding the equilibrium point where supply and demand curves intersect.
• Computer Graphics: Calculating where objects collide or overlap.
• Navigation: Pinpointing locations based on multiple bearings.

So, knowing how to solve these problems is a valuable skill! We will explore different methods to solve this problem, ensuring you understand the underlying concepts and can apply them to similar situations.

Method 1: Substitution Method

The substitution method is a powerful technique for solving systems of equations. The main idea is to solve one equation for one variable and then substitute that expression into the other equation. This will leave you with a single equation in one variable, which you can easily solve. Let's walk through the steps for our problem:

Step 1: Solve one equation for one variable.

Let's choose the second equation, x + y - 7 = 0, and solve for y. This looks straightforward since y has a coefficient of 1. Adding 7 and subtracting x from both sides gives us:

y = 7 - x

We've now expressed y in terms of x. This is a crucial step because we can now substitute this expression into the other equation.

Step 2: Substitute the expression into the other equation.

Now, we'll substitute our expression for y (7 - x) into the first equation, 2x - y + 4 = 0. Replacing y with (7 - x), we get:

2x - (7 - x) + 4 = 0

Notice that we've replaced the y with the entire expression we found in the previous step. It's essential to use parentheses here to ensure the negative sign is distributed correctly.

Step 3: Simplify and solve for x.

Now, let's simplify the equation and solve for x. Distribute the negative sign and combine like terms:

2x - 7 + x + 4 = 0 3x - 3 = 0

Add 3 to both sides:

3x = 3

Divide both sides by 3:

x = 1

Great! We've found the value of x. Now we're halfway there. Remember, the solution is a point, so we need both the x and y coordinates.

Step 4: Substitute the value of x back into either equation to solve for y.

We can use either of the original equations or the expression we found in Step 1 (y = 7 - x) to solve for y. The expression y = 7 - x is the easiest to use, so let's plug in x = 1:

y = 7 - 1 y = 6

Fantastic! We've found the value of y. Now we have both coordinates.

Step 5: Write the solution as a coordinate pair.

The solution is the point (x, y) = (1, 6). This is the point where the two lines intersect. It satisfies both equations, making it the solution to the system of equations.

Method 2: Elimination Method

The elimination method, also known as the addition method, is another powerful technique for solving systems of equations. The core idea is to manipulate the equations so that when you add them together, one of the variables cancels out (is eliminated). This leaves you with a single equation in one variable, which you can easily solve. Let's see how it works for our problem:

Step 1: Align the equations and make the coefficients of one variable opposites.

Our equations are:

2x - y + 4 = 0 x + y - 7 = 0

Notice that the y coefficients are already opposites (-1 and +1). This is perfect! If they weren't opposites, we would need to multiply one or both equations by a constant to make them opposites. For instance, if we had 2y in one equation and y in the other, we could multiply the second equation by -2 to get -2y.

Step 2: Add the equations together.

Now, we'll add the two equations together. This is where the magic happens:

(2x - y + 4) + (x + y - 7) = 0 + 0

Combining like terms, we get:

3x - 3 = 0

Notice that the y terms canceled out! This is exactly what we wanted. We're left with a single equation in x.

Step 3: Solve for x.

Now, let's solve for x. This is the same equation we got in the substitution method, so we know the drill:

3x - 3 = 0 3x = 3 x = 1

Excellent! We've found x = 1, just like we did with the substitution method. This gives us confidence that we're on the right track.

Step 4: Substitute the value of x back into either equation to solve for y.

Just like in the substitution method, we can plug x = 1 into either of the original equations to solve for y. Let's use the second equation, x + y - 7 = 0:

1 + y - 7 = 0 y - 6 = 0 y = 6

Fantastic! We've found y = 6, which is the same value we got using the substitution method.

Step 5: Write the solution as a coordinate pair.

The solution is the point (x, y) = (1, 6). This is the same answer we got using the substitution method, confirming that both methods lead to the correct solution.

Verification: Plugging the Solution Back In

It's always a good idea to verify your solution, especially in math problems. This ensures you haven't made any mistakes along the way. To verify our solution (1, 6), we'll plug these values back into both original equations and see if they hold true.

Equation 1: 2x - y + 4 = 0

Substitute x = 1 and y = 6:

2(1) - 6 + 4 = 0 2 - 6 + 4 = 0 0 = 0

This equation holds true! This is a good sign.

Equation 2: x + y - 7 = 0

Substitute x = 1 and y = 6:

1 + 6 - 7 = 0 7 - 7 = 0 0 = 0

This equation also holds true! Since our solution satisfies both equations, we can be confident that it's correct.

Visualizing the Solution: Graphing the Lines

Sometimes, a visual representation can help you understand the solution better. We can graph the two lines and see where they intersect. To graph the lines, we can rewrite the equations in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.

Equation 1: 2x - y + 4 = 0

Solve for y:

y = 2x + 4

This line has a slope of 2 and a y-intercept of 4.

Equation 2: x + y - 7 = 0

Solve for y:

y = -x + 7

This line has a slope of -1 and a y-intercept of 7.

If you were to graph these two lines, you would see that they intersect at the point (1, 6), which confirms our algebraic solution. Graphing is a fantastic way to double-check your work and gain a deeper understanding of the problem.

Conclusion: Mastering Linear Systems

Finding the point of intersection of two lines is a fundamental skill in algebra. We've explored two powerful methods – substitution and elimination – to solve this type of problem. Remember, the key is to manipulate the equations to isolate variables and ultimately find the (x, y) coordinates that satisfy both equations. By understanding these methods, you'll be well-equipped to tackle more complex problems in algebra and beyond.

So, guys, keep practicing, and you'll become masters of linear systems in no time! Happy solving!