Problem 64 - College Algebra and Calculus: An Applied Approach

Question

Reasoning Design a system of two linear equations with no solution. Solve the system algebraically and explain how the solution indicates that there is no solution.

Step-by-Step Solution

Answer

By creating the system of equations \(y = 2x + 3\) and \(y = 2x + 5\), and setting them equal to each other, the impossible solution \(3 = 5\) was reached, indicating that the original system of equations has no solution.

Step 1: Design the System of Equations

A simple pair of parallel lines can be represented by the equations \(y = 2x + 3\) and \(y = 2x + 5\).

Step 2: Set the Two Equations Equal to Each Other

To find the solution, one might try to set these two equations equal to each other, as such: \(2x + 3 = 2x + 5\).

Step 3: Solve the Equality

After simplifying, you would subtract \(2x\) from both sides to get \(3 = 5\). This is clearly not possible, and thus shows that the system of equations has no solution.

Key Concepts

Solving Linear Systems Algebraically

When we attempt to solve linear systems algebraically, we're essentially looking for a point, or a set of points, where the lines intersect. This means we are trying to find a common solution to all equations within the system.

Typically, methods like substitution or elimination are used to solve such systems. However, it's crucial to identify quickly whether the system has one solution, many solutions, or no solution. The equations given in the exercise, presented as parallel lines, indicate that algebraically we will not be able to consolidate them to find a common solution point.

For example, attempting to subtract one equation from the other to eliminate one variable will lead to a statement that is a contradiction (like asserting that 3 equals 5) and thereby proving algebraically that there is no solution.

Parallel Lines in Linear Equations

In geometry, parallel lines are defined as lines in a plane that never meet; they remain the same distance apart over their entire length.

Translating this concept into algebra, parallel lines correspond to linear equations that have the same slope but different y-intercepts. The slope of a line characterizes its steepness, and if two lines have equal slopes, they'll never intersect, just like parallel lines on a plane.

The exercise provided gives us two equations that graph to parallel lines. Since the lines do not intersect, no points exist that satisfy both equations simultaneously. This fundamental concept is why a system with equations representing parallel lines results in no solution.

No Solution in System of Equations

When we talk about a 'no solution' scenario in systems of equations, we mean that there are no set of values that simultaneously satisfy all equations in the system.

This situation occurs when the lines represented by the equations are parallel, as discussed before. No intersection point means no shared solution and this is presented, algebraically, as an equation with a false statement like the one obtained after Step 3 in the provided solution: '3 = 5'.

It's important for students to recognize these cases early in their algebraic reasoning to avoid unnecessary calculations and to understand the underlying relationships between the graphical and algebraic representations of systems of equations.

Algebraic Reasoning

Algebraic reasoning involves understanding and analyzing mathematical situations using algebraic methods and concepts. It's more than just executing calculations; it's about understanding what those calculations represent and what they can tell us about a given problem.

In this case, algebraic reasoning helps us to interpret the parallel nature of the lines without needing to graph them. By critically assessing the equations involved, students can predict that the system won't have a solution. Algebraic reasoning encompasses this ability to conceptualize mathematical relationships and to draw logical conclusions about these relationships.