Problem 64 - Precalculus with Limits

Question

Think About It Give examples of a system of linear equations that has (a) no solution and (b) an infinite number of solutions.

Step-by-Step Solution

Answer

An example of a system of linear equations with no solution could be \(y = 2x + 3\) and \(y = 2x - 1\). An example of a system with infinite solutions could be \(y = 3x + 2\) and \(y = 3x + 2\).

Step 1: Example of No solution

When we consider a system of linear equations with no solution, we are looking for two lines that are parallel and never intersect. One such example is the following:1. \(y=2x+3\)2. \(y=2x-1\)These two equations will never intersect since they have the same slope (2) but different y-intercepts (3 and -1), hence there is no common solution for x and y.

Step 2: Example of infinite solutions

A system of linear equations has infinite solutions if they are identical, meaning their slopes and y-intercepts are equal. For instance:1. \(y=3x+2\)2. \(y=3x+2\)Both equations represent the same line, so every point on that line is a solution, resulting in infinite solutions for x and y.

Key Concepts

No Solution

In systems of linear equations, a scenario of "No Solution" occurs when the lines are parallel. Parallel lines will never meet, no matter how far they are extended. This happens because they have identical slopes but different y-intercepts.
Let's consider an example:

• Equation 1: \( y = 2x + 3 \)
• Equation 2: \( y = 2x - 1 \)

Both equations have a slope of 2, meaning they rise and run in the exact same way. However, their y-intercepts differ (3 and -1), meaning they start at different points on the y-axis. Therefore, since they move in the same direction without meeting, there is no solution to this system.

Infinite Solutions

"Infinite Solutions" in a system of linear equations means that the two lines are actually the same line. This occurs when both the slope and the y-intercept of the equations are identical.
Here’s an example to illustrate:

• Equation 1: \( y = 3x + 2 \)
• Equation 2: \( y = 3x + 2 \)

In this case, each line has a slope of 3 and a y-intercept of 2. Since both lines lie perfectly on top of each other, they share all their points. This overlap means every point on the line is a solution, translating to an infinite number of solutions.

Parallel Lines

When we talk about parallel lines in the context of linear equations, we refer to lines that never meet. The key characteristic of parallel lines is that they constantly maintain an equal distance from each other. They have the same slope, which determines their steepness, making them rise and fall in unison.
However, parallel lines have different y-intercepts, meaning that they start at different points along the y-axis.

• For example: \( y = 4x + 5 \) and \( y = 4x - 3 \) show parallel lines.
• Both lines have a slope of 4 but different starting points (5 and -3).

Parallel lines illustrate a situation where no solution exists because they never cross each other, leaving no common point of intersection.

Identical Equations

Identical equations in a system of linear equations mean precisely what the term suggests: the equations are the same. This is often a giveaway for recognizing a system with infinite solutions. When both equations describe the same line, every \(x, y\) pair that lies on this line is a solution to the system.
Consider the following example:

• Equation 1: \( y = 5x + 7 \)
• Equation 2: \( y = 5x + 7 \)

Both equations are identical, having the same slope of 5 and a y-intercept of 7. Therefore, every point that satisfies Equation 1 will also satisfy Equation 2, leading to an infinite number of solutions. Identical equations tell us that instead of two lines, there is just one line described in two different ways.