Problem 54 - Algebra and Trigonometry

Question

Write the linear system whose solution set is \(\varnothing .\) Express each equation in the system in slope-intercept form.

Step-by-Step Solution

Answer

The system of linear equations with no solution can be \(y = 2x + 1\) and \(y = 2x + 3\). Both equations are in slope-intercept form.

Step 1: Identify meaning of no solution

A system of linear equations with no solution is one where the lines do not intersect at any point. One such example can be two parallel lines. They have the exact same slope but different y-intercepts.

Step 2: Write first equation in slope-intercept form

Choose a slope and a y-intercept for the first line. For instance, consider the slope \(m = 2\), and y-intercept \(b = 1\). Then the equation of the first line in slope-intercept form becomes \(y = 2x + 1\).

Step 3: Write second equation in slope-intercept form

Next, for the second line, choose the same slope as the first line to ensure they are parallel, but a different y-intercept. For example, let the y-intercept be \(b = 3\). So the equation for the second line becomes \(y = 2x + 3\) .

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a line is crucial for analyzing linear equations. It is expressed as \(y = mx + b\), where \(m\) represents the slope, and \(b\) is the y-intercept. The slope defines the steepness and direction of the line, while the y-intercept is the point where the line crosses the y-axis. To visualize this, imagine climbing up a hill: the slope describes how steep the hill is, and the y-intercept is where you started climbing from the flat ground.

To write an equation in slope-intercept form, we find these two key pieces of information. For example, if we're told a line has a slope \(m = 2\) and crosses the y-axis at \(1\), the equation would be \(y = 2x + 1\). This equation is handy because it immediately tells us how the line behaves and where it's positioned on a graph, easing the process of drawing the line or predicting the points it passes through.

Parallel Lines

Parallel lines share a fascinating attribute: they never meet, no matter how far they're extended. In the context of a coordinate plane, this means they have identical slopes. When two lines are parallel, you can almost think of them as train tracks running alongside each other, maintaining a constant distance apart.

In algebra, if we say two lines are parallel and we're using the slope-intercept form, the equations of the lines would have the same slope \(m\) but different y-intercepts \(b\). This is illustrated when we create a system of equations with no solution; we deliberately choose the same \(m\) for both lines, ensuring they stay parallel. For instance, if we have one line as \(y = 2x + 1\), to make another line parallel to it, we might choose \(y = 2x + 3\). They both rise and run in harmony, yet they will never intersect because their starting points (y-intercepts) are at different levels.

System of Linear Equations

When we work with more than one linear equation at the same time, we're dealing with a system of linear equations. Typically, we're trying to find a point that satisfies all the equations in the system - this is where the lines would intersect on a graph. However, systems can behave differently based on the relationships between their lines:

• If they intersect at one point, we say the system has a unique solution.
• If they lie on top of each other, we have infinitely many solutions.
• If they are parallel and do not intersect at all, the system has no solution, represented by the empty set \(\varnothing\).

A no-solution system sends a clear message: there's no single pair of \(x\) and \(y\) values that can satisfy both equations—just like two individual paths in life that never cross. It's an important concept that helps in understanding the nature of the lines we are dealing with and predicting the outcome of their interaction.