Problem 3 - Calculus for Biology and Medicine

Question

I n ~ P r o b l e m s ~ \(1-4\), solve each linear system of equations. In addition, for each system, graph the two lines corresponding to the two equations in a single coordinate system and use your graph to explain your solution. $$ \begin{aligned} x-2 y &=2 \\ y &=1+\frac{1}{2} x \end{aligned} $$

Step-by-Step Solution

Answer

The lines are parallel and do not intersect; the system has no solution.

Step 1: Understand the Equations

We have two linear equations: \( x - 2y = 2 \) and \( y = 1 + \frac{1}{2}x \). Our goal is to find the solution to this system of equations, which corresponds to the point where the two lines intersect.

Step 2: Rearrange the First Equation

For graphing purposes, let's rearrange the first equation \( x - 2y = 2 \) into the slope-intercept form (\( y = mx + b \)).\[x - 2y = 2 \Rightarrow -2y = -x + 2 \Rightarrow y = \frac{x}{2} - 1\]

Step 3: Plot the Equations

Using the slope-intercept forms, plot the two lines:- For \( y = \frac{x}{2} - 1 \), the slope (m) is \(\frac{1}{2}\) and the y-intercept is \(-1\).- For \( y = 1 + \frac{1}{2}x \), the slope (m) is \(\frac{1}{2}\) and the y-intercept is \(1\).Sketch these lines on the same graph.

Step 4: Identify the Intersection Point

By examining the graph, observe that both lines are parallel. They have identical slopes of \(\frac{1}{2}\) but different y-intercepts (\(-1\) and \(1\)). As such, they never intersect.

Step 5: Conceptual Explanation

Since there is no intersection point, the system of equations is considered inconsistent, meaning there is no solution. The parallel lines indicate that the system has no common solutions.

Key Concepts

Graphing Linear Equations

When graphing linear equations, you want to visualize their graphical representation. Each equation represents a straight line. Lines are infinitely long and extend in both directions. What's key is to correctly plot these lines on a graph so we can analyze their characteristics.

• The first step involves rewriting the equations in a suitable form, typically the slope-intercept form. This makes it easier to plot.
• Then, use the slope and y-intercept from the slope-intercept form to help map out the graph.

Visualizing the lines helps in understanding their intersections, if any. In cases where two lines intersect, the intersection point gives the solution to the system of equations. However, if they don't intersect, like parallel lines, there isn't a solution.

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most straightforward ways to express linear equations. It's given by: \[ y = mx + b \]Here:

• \( m \) is the slope of the line, indicating how steep the line is. A larger absolute value indicates a steeper line.
• \( b \) is the y-intercept, representing the point where the line crosses the y-axis.

This format simplifies plotting because it directly tells you two key pieces of information needed for easy graphing:

• The y-intercept point on the graph.
• The incline or decline direction and steepness.

Converting your equations to this form allows you to efficiently graph and examine relationships between multiple lines.

Parallel Lines

Parallel lines are lines in the same plane that never intersect. For two lines to be parallel, they must have the same slope but different y-intercepts.

• In our example, both lines had a slope of \( \frac{1}{2} \), which means they remain at a consistent distance apart.
• One line starts at a y-intercept of -1 and the other at 1.

The presence of parallel lines in a graph of a system of equations signifies something important: these lines never touch each other and thus don’t share any solutions. Understanding when lines are parallel helps quickly identify systems without solutions, known as inconsistent systems.

Inconsistent Systems

An inconsistent system is one that has no solutions. This arises when there is no point that satisfies all equations in the system.

• One of the most common scenarios leading to an inconsistent system is when the equations graph to parallel lines.
• Since parallel lines never intersect, there's no shared point or solution.

When dealing with practical problems, realizing a system is inconsistent can save time, avoiding unnecessary calculation. Recognizing the signs, such as parallel lines or having the same slope but different y-intercepts, clues us into this early. Understanding inconsistency is crucial in solving linear systems and interpreting their graphical depictions.