Solving Linear Equations By Graphing: A Step-by-Step Guide
Hey guys! Today, we're diving into solving a system of linear equations by graphing. It might sound intimidating, but trust me, it's super manageable once you break it down. We'll be tackling the following system:
Our main goal here is to figure out the solution to this system. So, let's get started!
Understanding Linear Equations and Graphing
Before we jump into the solution, let's quickly recap what linear equations are and why graphing is a useful method. Linear equations are equations that, when graphed, produce a straight line. They typically look like y = mx + b, where 'm' represents the slope and 'b' represents the y-intercept. Graphing is a visual way to solve systems of equations because the solution is simply the point where the lines intersect. If the lines never cross, there's no solution. If they are the same line, there are infinitely many solutions. This visual approach can make understanding the relationships between equations much easier.
When you're dealing with systems of linear equations, you're essentially looking for the values of x and y that satisfy both equations simultaneously. Think of it like finding a common ground β a point that works for both lines. Now, why graph? Graphing these equations helps us visualize this common ground. The point where the two lines intersect is the solution, because at that point, both equations hold true. This method is particularly helpful because it provides a clear visual representation of the solution. Itβs like having a map that guides you directly to the answer.
Graphing also helps in understanding different scenarios. Sometimes, the lines might not intersect at all, indicating that there's no solution to the system. Other times, the lines might overlap completely, meaning there are infinitely many solutions. By graphing the equations, we can quickly identify these situations. It provides a quick check that algebraic solutions might miss. For instance, if you solve algebraically and get a contradictory result (like 2=3), graphing would visually confirm that the lines are parallel and do not intersect. It makes the entire process more intuitive and less prone to errors. So, with that foundation in place, let's move on to solving our specific system.
Step 1: Graph the First Equation: $y=-\frac{5}{2}x-7$
Okay, let's tackle the first equation: $y=-\frac{5}{2}x-7$. To graph this, we'll use the slope-intercept form, which is y = mx + b. Remember, 'm' is the slope and 'b' is the y-intercept. In our equation, the slope (m) is -5/2, and the y-intercept (b) is -7. The y-intercept is where the line crosses the y-axis, so we can immediately plot the point (0, -7) on our graph. That's our starting point.
Now, let's use the slope to find another point. The slope, -5/2, tells us how much the line rises or falls for every unit we move to the right. A slope of -5/2 means that for every 2 units we move to the right on the x-axis, we move 5 units down on the y-axis. Starting from our y-intercept (0, -7), we move 2 units to the right and 5 units down. This brings us to the point (2, -12). Plot this point on the graph. You can continue this pattern to find more points if you like, but two points are really all you need to draw a straight line.
Once we have these two points, (0, -7) and (2, -12), we grab a ruler or straightedge and draw a straight line through them. Make sure the line extends beyond these points on both ends of the graph. This line represents all the possible solutions for the equation $y=-\frac{5}{2}x-7$. Think of every point on this line as a pair of x and y values that make the equation true. But remember, we're looking for the solution that works for both equations, so we're only halfway there. We've got one line; now we need to graph the second equation and see where they meet.
Step 2: Graph the Second Equation: $x + 2y = 4$
Alright, let's move on to the second equation: $x + 2y = 4$. This one looks a little different from the first, but don't worry, we can easily get it into the slope-intercept form (y = mx + b) by rearranging the terms. Our goal here is to isolate y on one side of the equation. So, let's start by subtracting x from both sides. This gives us:
Now, to completely isolate y, we divide both sides of the equation by 2:
Great! Now our equation is in the familiar slope-intercept form. We can see that the slope (m) is -1/2, and the y-intercept (b) is 2. This means the line crosses the y-axis at the point (0, 2). Let's plot that on our graph. This point is our new starting point for this line.
Next, we'll use the slope to find another point. A slope of -1/2 tells us that for every 2 units we move to the right on the x-axis, we move 1 unit down on the y-axis. Starting from the y-intercept (0, 2), we move 2 units to the right and 1 unit down. This lands us at the point (2, 1). Plot this point as well. Just like before, two points are enough to define our line, so we're good to go.
Now, grab that ruler or straightedge again, and draw a straight line through the points (0, 2) and (2, 1). Extend the line across the graph. This line represents all the solutions for the equation $x + 2y = 4$. We've now graphed both equations on the same coordinate plane. The magic happens where these two lines meet because that's the point that satisfies both equations simultaneously. So, let's find that intersection!
Step 3: Identify the Intersection Point
Okay, we've graphed both lines: $y=-\frac{5}{2}x-7$ and $y = -\frac{1}{2}x + 2$. Now comes the exciting part: identifying where these lines intersect. The intersection point is the solution to our system of equations because it's the only point that lies on both lines, meaning it satisfies both equations simultaneously.
Visually, on our graph, the intersection point is where the two lines cross each other. Take a close look at your graph, and pinpoint the exact coordinates of this point. It might fall neatly on grid lines, making it easy to read the x and y values. In this case, if you've drawn your graph accurately, you should see that the lines intersect at the point (-4, 3). This means that x = -4 and y = 3.
But before we declare victory, it's always a good idea to verify our solution. We can do this by plugging the x and y values into both original equations to make sure they hold true. Itβs like double-checking our map to ensure weβve reached the correct destination. This step ensures that we haven't made any mistakes in our graphing or reading of the graph.
So, let's plug x = -4 and y = 3 into our equations:
For the first equation, $y=-\frac{5}{2}x-7$:
That checks out! Now, let's try the second equation, $x + 2y = 4$:
Whoops! Something went wrong here. The equation does not hold true. It looks like we might have misread the graph or made a small error in our calculations or graphing. This is exactly why verifying the solution is so crucial. Let's go back and carefully re-examine our graph and our steps to pinpoint the mistake.
Step 4: Verify the Solution (and Correct if Necessary)
Okay, so we hit a snag in the verification step. This isn't a setback; it's actually a fantastic learning opportunity! It highlights why verifying your solution is so important. When we plugged x = -4 and y = 3 into the second equation, it didn't hold true. This tells us we need to revisit our graph and our calculations to find where we went wrong.
Let's start by re-examining the graph. Sometimes, the intersection point might appear to be at a certain coordinate, but a slight inaccuracy in our lines can throw us off. Looking closely, it seems there was a small error in plotting the first line. Let's recalculate and redraw that line with extra care.
The first equation is $y=-\frac{5}{2}x-7$. We plotted the y-intercept correctly at (0, -7). But when using the slope of -5/2, we need to make sure we move exactly 2 units to the right and 5 units down. If we do this precisely, we'll find that the line passes through the point (-2, -2) instead of (2, -12) as initially considered in the explanation. The initial calculation for the second point was incorrect. When you move 2 units to the right from (0,-7), you arrive at x=2. Then, moving 5 units down from -7 on the y-axis gets you to -12. Thus, the point (2, -12) is correct based on the slope and y-intercept, so there isn't necessarily an error in the slope calculation itself, but perhaps in visualizing and plotting this accurately on the coordinate plane.
The second equation is $x + 2y = 4$, which we converted to slope-intercept form as $y = -\frac{1}{2}x + 2$. The y-intercept is (0, 2), and the slope is -1/2. This means we move 2 units to the right and 1 unit down to find another point, which gives us the point (2, 1). This seems correct.
Now, letβs plot these points accurately and draw the lines again. If we do this carefully, we'll see that the lines actually intersect at the point (-4, 3). So our initial visual reading of the graph might have been correct, but let's revisit the verification step with a clearer graph.
Plugging x = -4 and y = 3 into the first equation:
That still checks out. Now, the second equation, $x + 2y = 4$:
Ah, we see the issue! There was an arithmetic error in the initial verification of the second equation. -4 + 6 does indeed equal 2, but it does not equal 4. Thus, the point (-4, 3) does not satisfy the second equation. Let's keep searching for the correct solution by more closely inspecting our graph.
By carefully re-plotting the lines and their points, we can more accurately determine the intersection. On closer inspection, it seems the intersection point is not exactly at integer coordinates. So, instead of relying solely on the graph, let's use an algebraic approach to solve the system and verify our graphical solution.
Step 5: Solve Algebraically to Confirm
Since our graphical method led to some ambiguity, letβs use an algebraic approach to find the precise solution. Weβll use the substitution method, as we already have the first equation solved for y:
And the second equation:
Substitute the first equation into the second:
Now, distribute and simplify:
Combine like terms:
Add 14 to both sides:
Divide by -4:
Now that we have x, plug it back into the first equation to find y:
So, the solution is $x = -\frac{9}{2}$ and $y = \frac{17}{4}$.
Step 6: Final Answer and Conclusion
Alright guys, we've made it! We initially set out to solve the system of linear equations by graphing, and we learned a valuable lesson about the importance of accuracy and verification along the way. While our initial attempt to read the solution directly from the graph had some hiccups, it pushed us to use a combination of graphical and algebraic methods for a more robust solution.
Our final solution, obtained through the algebraic substitution method, is:
So, the point of intersection for the two lines is $(-4.5, 4.25)$.
In conclusion, solving systems of linear equations by graphing is a powerful visual tool, but it's essential to be precise and always verify your solution. When graphical methods become ambiguous, remember that algebraic methods are there to provide a definitive answer. The combination of both approaches gives us a solid understanding and accurate solution. Keep practicing, and you'll become a pro at solving these problems!