Solving Muriel's Equation For Infinite Solutions

In the fascinating realm of linear equations, a system with an infinite number of solutions presents a unique and intriguing scenario. This article delves into the concept of such systems, specifically focusing on Muriel's equation and the quest to identify a second equation that, when paired with the first, results in an infinite solution set. We will explore the underlying principles, analyze the given options, and ultimately unravel the mystery behind Muriel's equation. Prepare to embark on a journey through the world of linear equations, where infinite possibilities await!

Understanding Systems with Infinite Solutions

When we talk about systems of linear equations having an infinite number of solutions, we're essentially saying that the two equations represent the same line. This might sound a bit odd at first, but let's break it down. A linear equation in two variables, like x and y, geometrically represents a straight line. When we have two linear equations, we're dealing with two lines on a graph. The solution to the system of equations is the point (or points) where these lines intersect. Now, if the two lines are actually the same line, they intersect at every point along their length. This means there are infinitely many points of intersection, and thus, infinitely many solutions.

Key Concept: For two linear equations to represent the same line, they must be scalar multiples of each other. This means that one equation can be obtained by multiplying the other equation by a constant. This is the core principle we'll use to solve Muriel's equation.

To identify equations that are scalar multiples, we need to express them in a standard form, such as slope-intercept form (y = mx + b) or standard form (Ax + By = C). This allows us to easily compare the coefficients and constants and determine if a proportional relationship exists. For instance, if one equation is 2x + 4y = 6, an equation representing the same line could be obtained by multiplying the entire equation by 2, resulting in 4x + 8y = 12. Similarly, dividing the original equation by 2 would yield x + 2y = 3, which also represents the same line. The critical aspect is that the ratio between the coefficients of x, y, and the constant term remains consistent across equivalent equations.

Consider the equation 3x - 6y = 9. If we divide the entire equation by 3, we get x - 2y = 3. Both equations represent the same line and thus have infinitely many solutions when considered as a system. Alternatively, if we multiply the original equation by -1, we obtain -3x + 6y = -9, which again represents the same line. Recognizing these scalar multiples is paramount in determining systems with infinite solutions. Furthermore, understanding this concept has practical implications in various fields, such as economics and engineering, where systems of equations are used to model and solve real-world problems. In economics, for example, supply and demand curves may overlap infinitely under specific conditions, leading to market equilibrium scenarios with numerous possible solutions. In engineering, structural designs sometimes involve systems of equations that must be carefully analyzed to avoid infinite solutions, which could indicate instability or redundancy in the design.

Muriel's Equation: The Challenge

Muriel has presented us with a fascinating puzzle. She has written a system of two linear equations with an infinite number of solutions, and one of those equations is: 3y = 2x - 9. Our mission is to find the other equation from the given options that, when paired with this equation, creates a system with infinitely many solutions. This means we need to find an equation that is a scalar multiple of 3y = 2x - 9. To make this task easier, let's first rewrite Muriel's equation in slope-intercept form (y = mx + b) by dividing both sides by 3:

y = (2/3)x - 3

Now we have Muriel's equation in a more recognizable form, where the slope (m) is 2/3 and the y-intercept (b) is -3. Any equation that represents the same line will have the same slope and y-intercept, or be a scalar multiple of the equation in standard form. This transformation is a crucial step because it allows us to directly compare the coefficients and constants of the given options with Muriel's equation. By converting the equation into slope-intercept form, we've essentially created a template against which we can measure the other potential equations. This approach is not only efficient but also provides a clear visual representation of the line, making it easier to identify equivalent equations. Moreover, this technique is widely applicable in various mathematical and real-world scenarios where comparing and analyzing linear relationships is essential.

The slope-intercept form is particularly useful because it directly reveals the line's key characteristics: its steepness (slope) and where it crosses the y-axis (y-intercept). These characteristics are unique to each line, and if two lines share the same slope and y-intercept, they are essentially the same line, just represented in potentially different forms. Therefore, our task now is to examine the given options and determine which one, when manipulated, can be transformed into the same slope-intercept form as Muriel's equation or a scalar multiple of its standard form. This methodical approach ensures that we consider all aspects of the equation and accurately identify the one that represents the same line, thus providing the infinite solutions we're seeking.

Analyzing the Options

Now, let's meticulously examine the provided options to determine which one, when paired with Muriel's equation, will result in an infinite number of solutions. Remember, we're looking for an equation that is a scalar multiple of y = (2/3)x - 3.

Option 1: 2y = x - 4.5

To analyze this, let's convert it to slope-intercept form by dividing both sides by 2:

y = (1/2)x - 2.25

Comparing this to Muriel's equation, y = (2/3)x - 3, we see that the slopes (1/2 and 2/3) and the y-intercepts (-2.25 and -3) are different. Therefore, this equation represents a different line and will not result in an infinite number of solutions. The discrepancy in the slopes indicates that these lines will intersect at a single point, representing a unique solution, not an infinite set. Furthermore, the different y-intercepts confirm that the lines cross the y-axis at distinct points, solidifying the conclusion that they are not the same line. Thus, Option 1 can be confidently ruled out as a potential solution to Muriel's equation puzzle.

Option 2: y = (2/3)x - 3

This equation is identical to Muriel's equation after we rewrote it in slope-intercept form! This means they represent the same line. Therefore, this equation, when paired with Muriel's equation, will result in an infinite number of solutions. The matching slopes and y-intercepts leave no doubt that these equations describe the same line, thus fulfilling the condition for infinite solutions. This outcome underscores the core principle of linear systems with infinite solutions: the equations must be scalar multiples or, in this case, identical.

Option 3: 6y = 4x - 18

Let's convert this equation to slope-intercept form by dividing both sides by 6:

y = (4/6)x - 3

Simplifying the fraction, we get:

y = (2/3)x - 3

This equation is also identical to Muriel's equation! This confirms that it represents the same line and will result in an infinite number of solutions when paired with Muriel's equation. This option provides an additional confirmation of the principle of scalar multiples. By dividing the equation by 6, we've effectively scaled it down to match Muriel's equation, demonstrating that they are indeed equivalent representations of the same line. This reinforces the understanding that infinite solutions arise when the equations in a system are essentially different forms of the same underlying linear relationship.

The Solution: Infinite Possibilities

After carefully analyzing the options, we've discovered that Option 2 (y = (2/3)x - 3) and Option 3 (6y = 4x - 18) both represent the same line as Muriel's equation (3y = 2x - 9). Therefore, either of these equations, when paired with Muriel's equation, will create a system with an infinite number of solutions. This means that there isn't just one correct answer, but rather two, highlighting the multiple ways in which the same linear relationship can be expressed. This insight is crucial in understanding the flexibility and interconnectedness of linear equations.

The key to solving this problem was recognizing that equations representing the same line are scalar multiples of each other. By converting Muriel's equation and the options into slope-intercept form, we were able to easily compare their slopes and y-intercepts and identify the equivalent equations. This approach not only provided the solution but also reinforced the fundamental concepts of linear systems and their graphical representation. Furthermore, the exercise demonstrates the practical importance of manipulating equations into standard forms to facilitate comparison and analysis. In more complex mathematical problems, this skill is invaluable in simplifying expressions and revealing underlying relationships.

In conclusion, Muriel's equation puzzle serves as an excellent illustration of how systems of linear equations can have infinite solutions when the equations represent the same line. The process of identifying these equations involves a careful examination of their slopes, y-intercepts, and scalar relationships. By mastering these concepts, we can confidently navigate the world of linear equations and solve a wide range of problems, both theoretical and practical. The ability to recognize equivalent equations and understand their implications is a fundamental skill in mathematics and has broad applications across various scientific and engineering disciplines.