Solving Linear Equations: Slope-Intercept & Consistency
Hey guys! Today, we're diving into the world of linear equations, focusing on how to graph them using the slope-intercept form and how to determine if a system of equations is consistent or inconsistent. If a system is consistent, we'll also find its solution. Let's break it down step-by-step!
Graphing Linear Equations in Slope-Intercept Form
First, let's tackle graphing the linear equations by understanding the slope-intercept form. The slope-intercept form of a linear equation is given by y = mx + b, where m represents the slope and b represents the y-intercept. When you have equations in this form, graphing becomes super easy. However, sometimes you'll encounter equations that don't immediately look like y = mx + b. In those cases, you'll need to rearrange the equation to fit this form. Understanding this form is crucial because the slope m tells you how steep the line is and in what direction it's going (uphill or downhill), and the y-intercept b tells you where the line crosses the y-axis. Knowing these two things makes plotting the line straightforward. Think of the y-intercept as your starting point on the y-axis and the slope as the instruction manual for how to move from that point to draw the rest of the line. Whether you are dealing with simple or complex equations, mastering slope-intercept form gives you a powerful tool for visualizing and understanding linear relationships. The ability to quickly convert equations to slope-intercept form allows you to immediately grasp key features of the line. This skill is invaluable not only for graphing but also for solving systems of equations, analyzing data, and understanding various real-world phenomena that can be modeled linearly. Grasping this foundational concept ensures you can confidently tackle more advanced topics in algebra and beyond. For instance, if you have an equation like 2y = 4x + 6, you'd divide everything by 2 to get y = 2x + 3, revealing a slope of 2 and a y-intercept of 3. This makes it clear where to start plotting your line and how to continue it across the graph. Let's apply this to the given equations:
Equation 1: y = 5
This equation is already in a simplified form! We can think of it as y = 0x + 5. This means the slope (m) is 0, and the y-intercept (b) is 5. To graph this, you simply draw a horizontal line that passes through the point (0, 5) on the y-axis. It’s a flat line because the y value is always 5, no matter what the x value is. Think of it like a constant height – no uphill or downhill, just a straight, unwavering path across the graph.
Equation 2: x = -17
This equation represents a vertical line. No matter what the y value is, x is always -17. To graph this, you draw a vertical line that passes through the point (-17, 0) on the x-axis. It’s a straight up-and-down line because the x value is always -17, regardless of the y value. Picture it as a fixed position on the x-axis – it never moves left or right, only up and down.
Determining Consistency and Finding Solutions
Now, let's determine whether the system of equations is consistent or inconsistent. A consistent system is a system of equations that has at least one solution. This means the lines intersect at one or more points. An inconsistent system is a system of equations that has no solution. This means the lines are parallel and never intersect.
Analyzing the System
We have two equations:
- y = 5 (a horizontal line)
- x = -17 (a vertical line)
Since a horizontal line and a vertical line will always intersect at one point, this system is consistent. They are not parallel; they meet at a single, distinct location. The beauty of having one horizontal and one vertical line is that their intersection is guaranteed unless we were dealing with parallel lines, which isn't the case here. This makes it much easier to determine whether the system has a solution. In our example, the equations y = 5 and x = -17 couldn't be more different in their orientation, which assures us they must cross paths somewhere on the coordinate plane. Understanding the graphical representation of each equation type is key to rapidly assessing system consistency. Consistent systems are the foundation for many mathematical models and real-world applications. When you encounter situations where multiple conditions must be satisfied simultaneously, you are often looking at a system of equations. This is especially relevant in fields like engineering, economics, and computer science, where interrelated variables need to be precisely coordinated. Knowing whether a solution exists and being able to find it is crucial for problem-solving in these domains.
Finding the Solution
The solution to the system is the point where the two lines intersect. In this case, the horizontal line y = 5 intersects the vertical line x = -17 at the point (-17, 5). This is where both equations are true simultaneously. To confirm, you can substitute these values back into the original equations:
- For y = 5: 5 = 5 (True)
- For x = -17: -17 = -17 (True)
So, the solution is indeed (-17, 5).
Conclusion
In summary, we've graphed the linear equations y = 5 and x = -17. We determined that the system is consistent and found the solution to be (-17, 5). Understanding how to convert equations into slope-intercept form and recognizing that horizontal and vertical lines intersect is key to solving these types of problems. Keep practicing, and you'll master these concepts in no time!
So, the answer is:
A. Consistent
