Graph Of Y Equals -4x Plus 7 Explained

The equation y = -4x + 7 represents a fundamental concept in algebra: a linear equation. This article will delve deep into understanding the graph of this equation, option A, and its significance in representing solutions. We will explore why the graph is a line, how it embodies all solutions, and contrast it with other options to provide a comprehensive understanding. We'll also touch upon related concepts like slope, y-intercept, and how to graph linear equations, ensuring a thorough grasp of this essential mathematical concept.

Option A: A Line That Shows the Set of All Solutions to the Equation

This is the correct answer. The graph of the equation y = -4x + 7 is indeed a straight line. But what does this line represent? It represents the set of all possible solutions to the equation. Each point on the line corresponds to an ordered pair (x, y) that, when substituted into the equation, makes the equation true. This is a crucial concept in understanding linear equations and their graphical representations. To truly appreciate this, let's break down why this is the case and explore the characteristics of a linear equation.

A linear equation, in its simplest form, is an equation that can be written in the form y = mx + b, where 'm' represents the slope of the line and 'b' represents the y-intercept. In our case, y = -4x + 7, we can clearly see that m = -4 and b = 7. The slope, -4, tells us the steepness and direction of the line. It signifies that for every one-unit increase in x, y decreases by four units. This consistent rate of change is what gives the line its straight form. The y-intercept, 7, is the point where the line intersects the y-axis. This is the point where x = 0, and the y-value is 7. So, the point (0, 7) lies on the line.

Now, consider any point on this line. Let's say we choose x = 1. Substituting this into the equation, we get y = -4(1) + 7 = 3. So, the point (1, 3) should lie on the line. If we choose x = 2, we get y = -4(2) + 7 = -1. So, the point (2, -1) should also lie on the line. We can continue this process indefinitely, choosing different values for x and calculating the corresponding y-values. Each pair (x, y) that we obtain will satisfy the equation, and each will represent a point on the line. This is why the line represents the set of all solutions – because every point on the line is a solution to the equation, and every solution to the equation corresponds to a point on the line.

This concept is fundamental to solving systems of linear equations. When you have two linear equations, the solution to the system is the point where their lines intersect. This point represents the (x, y) values that satisfy both equations simultaneously. Understanding that a line represents all solutions to an equation is also crucial in linear programming, where we seek to optimize a linear objective function subject to linear constraints. The feasible region in linear programming is often defined by a system of linear inequalities, and the boundary of this region is composed of lines that represent the equations corresponding to these inequalities.

In summary, the graph of y = -4x + 7 is a line because the equation is linear. This line embodies all possible solutions because every point on the line represents an (x, y) pair that satisfies the equation. This understanding is crucial for grasping more advanced concepts in algebra and calculus.

Why Other Options are Incorrect

To fully grasp the correctness of option A, it's important to understand why the other options are incorrect. This will solidify your understanding of linear equations and their graphical representations.

Option B: A Point That Shows the y-intercept

While the y-intercept is indeed an important characteristic of the line, it is just one point on the line. The y-intercept is the point where the line crosses the y-axis, and in the equation y = -4x + 7, the y-intercept is (0, 7). However, a single point cannot represent all the solutions to the equation. As we discussed in detail earlier, a linear equation has an infinite number of solutions, each represented by a point on the line. Therefore, while the y-intercept is a significant point, it doesn't encompass the entirety of the solution set.

Think of it this way: the y-intercept is like the starting point of the line on the y-axis, but it doesn't tell us anything about the line's direction or how it extends in both directions. The line continues infinitely in both directions, passing through countless other points, each representing a different solution. Focusing solely on the y-intercept would be like trying to understand an entire road trip by only looking at the starting location – you'd miss the vast majority of the journey.

Option C: A Point That Shows One Solution to the Equation

This option is partially correct but incomplete. As we've established, each point on the line represents one solution to the equation. However, the graph isn't just a single point; it's a collection of infinitely many points, each representing a different solution. Choosing this option would be like saying a novel is just one word – technically correct, but a gross underrepresentation of the whole.

Consider the point (1, 3), which we previously calculated to be a solution. This point does represent one solution to the equation. However, it's just one out of countless possibilities. There are infinitely many other x-values we could choose, each resulting in a different y-value, and each (x, y) pair will be a solution represented by a point on the line. So, while a point can represent a single solution, it cannot represent the entire set of solutions.

Option D: A Line That Shows Only One Solution to the Equation

This option is fundamentally incorrect. A line, in the context of a linear equation, represents infinitely many solutions, not just one. This is the core concept we've been exploring. The very nature of a linear equation, with its constant slope and y-intercept, dictates that it will have an infinite number of solutions. Each point on the line is a solution, and there are infinitely many points on a line.

To reiterate, the line is the visual representation of all the (x, y) pairs that satisfy the equation. If we were to pick any x-value and plug it into the equation, we could solve for a corresponding y-value. This (x, y) pair would be a point on the line, and thus, a solution. Since we can pick infinitely many x-values, there are infinitely many solutions, and the line represents them all.

In conclusion, understanding why options B, C, and D are incorrect reinforces the understanding of why option A is the correct answer. The graph of a linear equation is not just a point or a single solution; it's a line that beautifully illustrates the infinite set of solutions to the equation.

Further Exploration: Slope, Y-intercept, and Graphing

To further solidify your understanding of the graph of y = -4x + 7, let's delve into the concepts of slope and y-intercept and briefly discuss how to graph the equation.

Slope

The slope of a line, often denoted by 'm', is a measure of its steepness and direction. In the equation y = mx + b, 'm' is the slope. In our case, the slope is -4. This means that for every one-unit increase in x, the value of y decreases by four units. A negative slope indicates that the line is decreasing as you move from left to right. In other words, the line slopes downwards.

The slope can also be interpreted as the "rise over run." A slope of -4 can be written as -4/1, meaning that for every 1 unit you "run" horizontally (increase in x), you "rise" -4 units vertically (decrease in y). This consistent ratio is what gives the line its straight form. Understanding the slope is crucial for interpreting the behavior of the linear equation and its graph. A steeper slope (a larger absolute value of m) indicates a more rapid change in y for a given change in x.

Y-intercept

The y-intercept, denoted by 'b' in the equation y = mx + b, is the point where the line intersects the y-axis. This is the point where x = 0. In the equation y = -4x + 7, the y-intercept is 7. This means that the line crosses the y-axis at the point (0, 7). The y-intercept provides a fixed reference point for the line's position on the coordinate plane.

The y-intercept is particularly useful when graphing a linear equation. It gives you one point on the line immediately. Combined with the slope, you can easily determine other points on the line and draw its graph. The y-intercept also has practical significance in real-world applications of linear equations. For example, if the equation represents the cost of a service based on usage, the y-intercept might represent the fixed cost or initial fee.

Graphing the Equation

There are several ways to graph the equation y = -4x + 7. One common method is to use the slope-intercept form (y = mx + b) and plot the y-intercept first. In our case, we know the y-intercept is (0, 7). Next, use the slope to find another point on the line. The slope is -4, which can be written as -4/1. This means we can start at the y-intercept (0, 7), move 1 unit to the right (increase x by 1), and then move 4 units down (decrease y by 4). This will give us the point (1, 3).

Now that we have two points, (0, 7) and (1, 3), we can draw a straight line through them. This line is the graph of the equation y = -4x + 7. You can also find additional points by substituting different values for x into the equation and solving for y. For example, if we substitute x = 2, we get y = -4(2) + 7 = -1, giving us the point (2, -1). This point should also lie on the line we've drawn.

Another method for graphing is to find the x-intercept, which is the point where the line crosses the x-axis (where y = 0). To find the x-intercept, set y = 0 in the equation and solve for x: 0 = -4x + 7. Solving for x, we get x = 7/4 = 1.75. So, the x-intercept is (1.75, 0). Now you have two points, the y-intercept (0, 7) and the x-intercept (1.75, 0), and you can draw a line through them.

Graphing linear equations is a fundamental skill in algebra. It allows you to visualize the relationship between the variables and understand the equation's solutions in a geometric context. Practice graphing different linear equations to develop your proficiency and intuition for this important concept.

Conclusion

The graph of the equation y = -4x + 7 is A. a line that shows the set of all solutions to the equation. This understanding is fundamental to grasping linear equations and their representations. We've explored why this is the case, contrasting it with other options and delving into the concepts of slope, y-intercept, and graphing. By understanding these principles, you build a solid foundation for more advanced mathematical concepts and problem-solving.