System Of Equations: Identical Lines Explained

Dec 11, 2025 by Alex Johnson 47 views

Let's dive into the fascinating world of systems of equations! Today, we're going to dissect a specific system:

$egin{array}{c} y=2 x+3 \ 4 x-2 y=-6 end{array}$

We'll be exploring the true statements about this system, focusing on what happens when we graph these equations and analyze their slopes. Understanding these concepts is crucial for mastering algebra and problem-solving. So, grab a cup of coffee, get comfy, and let's break it down!

The Graphing Method: A Visual Approach

The graphing method is a fantastic visual tool to understand systems of equations. It involves plotting each equation on a coordinate plane and observing where the lines intersect. The point(s) of intersection represent the solution(s) to the system. If the lines intersect at one point, there's a unique solution. If they are parallel and never intersect, there's no solution. But what happens when the lines are exactly the same? This is precisely the situation we'll explore with our given system. When the graphing method shows that the lines are the same, it means that every single point on one line is also on the other line. This implies that there isn't just one solution, but an infinite number of solutions, because every point on that line satisfies both equations. It's like having two descriptions that perfectly describe the same object – they are fundamentally identical. To confirm this visually, you would plot both equations, and instead of seeing two distinct lines, you would see only one single line because they overlap perfectly. This visual confirmation is powerful, and it directly leads us to understand the nature of the solutions for such systems. We can even algebraically manipulate the equations to see if they are indeed the same, which we will do later.

Analyzing the Slopes: Unveiling the Relationship

Now, let's talk about slopes. The slope of a line tells us its steepness and direction. For a linear equation in the form $y = mx + b$ , 'm' represents the slope and 'b' represents the y-intercept. In our system, the first equation is already in this form: $y = 2x + 3$ . Here, the slope ( $m_1$ ) is 2, and the y-intercept ( $b_1$ ) is 3.

The second equation is $4x - 2y = -6$ . To easily compare its slope and y-intercept, we need to convert it into the slope-intercept form ( $y = mx + b$ ). Let's do that:

Subtract $4x$ from both sides: $-2y = -4x - 6$
Divide both sides by -2: $y = \frac{-4x}{-2} - \frac{6}{-2}$ $y = 2x + 3$

Wow! Look at that! After rearranging, the second equation becomes identical to the first equation. This means that the slope of the second line ( $m_2$ ) is also 2, and its y-intercept ( $b_2$ ) is also 3.

When two lines have the same slope and the same y-intercept, they are not just parallel; they are identical. They are literally the same line. This is a key insight. If the slopes were different, the lines would intersect at a single point, giving a unique solution. If the slopes were the same but the y-intercepts were different, the lines would be parallel and never intersect, resulting in no solution. But in our case, everything matches perfectly.

Understanding the Solutions: Infinite Possibilities

Since both equations simplify to $y = 2x + 3$ , they represent the exact same line. This means that any point $(x, y)$ that satisfies the first equation will automatically satisfy the second equation, and vice versa. This is why such a system has infinite solutions. Every point on the line $y = 2x + 3$ is a valid solution to the system.

Think about it this way: if you were asked to find a number that is equal to itself, any number would work, right? That's similar to this system. We have two equations that are essentially saying the same thing. Therefore, any pair of $(x, y)$ values that fits the description $y = 2x + 3$ is a solution. We can express these infinite solutions using set notation:

$\{ (x, y) \mid y = 2x + 3 \}$

This notation reads as "the set of all points $(x, y)$ such that $y$ is equal to $2x + 3$ ." It's a concise way to represent all the points that lie on that single, continuous line. This concept of infinite solutions is a direct consequence of the two equations being dependent, meaning one can be derived from the other, and in this specific case, they are identical.

True Statements About the System

Based on our analysis, let's identify the true statements about the system:

The graphing method will show that the lines are the same line. As we saw, when we graph both equations, they will perfectly overlap, appearing as a single line. This visual representation confirms their identical nature.
The slopes of the lines are equal. Both lines have a slope of 2. This equality of slopes is a prerequisite for the lines to be either parallel or identical.
The y-intercepts of the lines are equal. Both lines have a y-intercept of 3. This, combined with equal slopes, confirms that the lines are identical.
The system has infinitely many solutions. Because the two equations represent the same line, every point on that line is a solution to the system.
The equations are dependent. One equation can be derived from the other (in this case, they are identical), meaning they don't provide independent pieces of information about the relationship between x and y. They are essentially two ways of stating the same fact.

When Lines Are the Same: A Deeper Look

When we encounter a system of linear equations where both equations simplify to the exact same form, like in our example ( $y = 2x + 3$ ), it signifies a special relationship between them. These are not just any lines; they are identical lines. This is a critical concept in algebra. It means that the two equations are not providing distinct information. Instead, they are redundant. You could think of it as having two identical keys that open the same lock – they both do the same job. Mathematically, this redundancy leads to the conclusion that there isn't a single, unique point where these lines intersect because they are, in fact, the same line. Therefore, every point that lies on this line satisfies both equations simultaneously. This leads to the conclusion of infinite solutions.

Consider the implications. If you were solving a real-world problem modeled by such a system, it would suggest that the constraints or conditions described by the two equations are not independent. For instance, if you had two different ways of describing the cost of a certain number of items, and both descriptions led to the same formula, it would mean that the second description didn't add any new information. This is why understanding dependent equations and identical lines is so important – it helps us interpret the results of our mathematical models accurately. The slopes being equal ( $m_1 = m_2$ ) and the y-intercepts being equal ( $b_1 = b_2$ ) are the direct indicators that we are dealing with identical lines and, consequently, infinite solutions. It’s a beautiful symmetry in mathematics that allows us to predict the nature of solutions just by examining the coefficients and constants in the equations.

Conclusion: The Beauty of Identical Lines

In summary, the system of equations

$egin{array}{c} y=2 x+3 \ 4 x-2 y=-6 end{array}$

boils down to a single, identical line: $y = 2x + 3$ . This means:

The graphing method visually confirms they are the same line.
The slopes are equal (both are 2).
The y-intercepts are equal (both are 3).
The system has infinitely many solutions.
The equations are dependent.

Understanding these characteristics is fundamental to solving and interpreting systems of linear equations. It allows us to distinguish between unique solutions, no solutions, and infinite solutions, providing a complete picture of the relationships between the variables.

For further exploration into systems of equations and linear algebra, I recommend visiting Khan Academy. They offer excellent resources and practice problems to deepen your understanding.