Problem 63 - College Algebra and Calculus: An Applied Approach

Question

Reasoning Design a system of two linear equations with infinitely many solutions. Solve the system algebraically and explain how the solution indicates that there are infinitely many solutions.

Step-by-Step Solution

Answer

We created the system of equations \(y = 2x + 3\) and \(2y - 4x = 6\). Solving this system algebraically, both equations reduce to the same form, \(y - 2x = 3\), indicating that there are infinitely many solutions.

Step 1: Design the System of Equations

Choose any line in a 2D space, such as \(y = 2x + 3\). To create a system with infinitely many solutions, we can multiply both sides of this equation by an arbitrary nonzero value to get the second equation. Let's multiply by 2 and we get \(2y = 4x + 6\). So, our system of equations is:System: \[\begin{{array}}{{l}}y = 2x + 3 \2y - 4x = 6 \end{{array}}\]

Step 2: Solve System Algebraically

In the second equation, divide all terms by 2 to get the equation in a similar form to the first one:\[\begin{{array}}{{l}}y - 2x = 3 \end{{array}}\]So, now the system of equations is:\[\begin{{array}}{{l}}y = 2x + 3 \y - 2x = 3 \end{{array}}\]These two equations are identical, showing that they will graph the same line.

Step 3: Explain the Solution

Because the lines are identical, they will intersect at every point along the line, which means there are infinitely many solutions. For every x value, the y value will be the same for both equations. Thus, every coordinate point (x, 2x + 3) is a solution.

Key Concepts

Infinite Solutions

Understanding the concept of infinite solutions is essential in solving systems of linear equations. When we say a system has infinite solutions, we mean that there isn't just a single set of values that satisfies all equations in the system. Instead, there is a whole set of different values, in this case, every single point along a line, which solves the system.

In the given exercise, the system was designed in such a way that both equations represent the same line. This situation occurs when all equations in the system are just multiplied versions of each other, or in other words, are proportional to each other. Therefore, instead of crossing at a single point, the lines lie on top of each other, resulting in them intersecting at every point along the line. This scenario symbolizes a key characteristic of systems with infinite solutions: the equations are dependent on one another, creating a single continuous solution set.

Algebraic Methods

The term algebraic methods refers to the techniques used to solve equations or systems of equations. It encompasses a variety of tools such as substitution, elimination, and graphing. For systems of linear equations, these methods aim to simplify the equations to find the values of the unknowns that satisfy all equations simultaneously.

The choice of method often depends on the nature of the system. In the exercise, simplification and observation were used to demonstrate that the system had infinitely many solutions. By manipulating the second equation to match the first, it becomes clear that the two equations are actually the same line described differently. This is a clever algebraic technique to transform the system into a simpler form, where the result becomes more obvious.

Solving Systems of Equations

The process of solving systems of equations involves finding the values for the unknown variables that make all equations true at the same time. When the system involves linear equations, it can result in a single unique solution, no solution, or infinitely many solutions.

A common approach is to graph the equations, look for intersections, and interpret the result. Alternatively, algebraic methods, such as the ones mentioned above, can provide a more precise solution. In the exercise, we saw that the system had infinite solutions because both equations were essentially the same. Understanding the types of outcomes can guide students on how to approach a system and predict the possible solutions before even solving it.

Linear Algebra

Linear algebra is a field of mathematics that focuses on vector spaces and linear mappings between these spaces. It includes the study of lines, planes, and subspaces, and is fundamental in modern mathematics and applications such as science, engineering, computer science, economics, and statistics.

Systems of linear equations are at the heart of linear algebra. They are usually represented in matrix form, which makes them amenable to a range of powerful solution methods, including matrix operations and factorizations. In the context of our exercise, linear algebra provides the theoretical foundation that allows us to understand why certain manipulations, like multiplying an equation by a constant, lead to equivalent expressions and hence, to a system with infinite solutions.