Problem 44 - College Algebra with Corequisite Support

Question

For the following exercises, graph the system of equations and state whether the system is consistent, inconsistent, or dependent and whether the system has one solution, no solution, or infinite solutions. $$ \begin{array}{c} 3 x-5 y=7 \\ x-2 y=3 \end{array} $$

Step-by-Step Solution

Answer

The system is consistent with one solution.

Step 1: Understand the System of Equations

We are given two equations: 1. \(3x - 5y = 7\) 2. \(x - 2y = 3\). The task is to graph these equations and determine the nature of their solution.

Step 2: Convert Equations to Slope-Intercept Form

To graph the equations, convert them into slope-intercept form, \(y = mx + b\).- For the first equation \(3x - 5y = 7\), solve for \(y\): \(-5y = -3x + 7\) or \(y = \frac{3}{5}x - \frac{7}{5}\).- For the second equation \(x - 2y = 3\), solve for \(y\): \(-2y = -x + 3\) or \(y = \frac{1}{2}x - \frac{3}{2}\).

Step 3: Graph Each Line

Using the slope-intercept forms, graph each equation:- Line 1: Start at \((0, -\frac{7}{5})\) with a slope of \(\frac{3}{5}\).- Line 2: Start at \((0, -\frac{3}{2})\) with a slope of \(\frac{1}{2}\).Plot these lines carefully on a graph, ensuring each line accurately reflects its respective slope and y-intercept.

Step 4: Determine Intersection Point

Look for the point where both lines intersect on the graph. This point represents the solution to the system of equations. If the lines intersect at a single point, the system has one solution.

Step 5: Analyze the Consistency and Dependency

Based on the graph: - If the lines intersect at one point, the system is consistent and has one solution. - If the lines are parallel and do not intersect, the system is inconsistent and has no solution. - If the lines coincide (are the same line), the system is dependent and has infinite solutions.

Key Concepts

Graphing Linear Equations

When you graph a linear equation, you represent it visually on the coordinate plane. Each equation results in a straight line. To graph a linear equation, it's often helpful to convert it into slope-intercept form (more on that later).

• Select a few values for x, substitute these into the equation, and solve for y, this gives points on the line.
• Plot these points on the graph, and connect them with a straight line.

Graphing two linear equations on the same set of axes can help you understand how the two interact. It's important to do this accurately, particularly when identifying where lines might intersect.

Slope-Intercept Form

The slope-intercept form of a linear equation is expressed as \(y = mx + b\). This form is particularly helpful for graphing because it clearly indicates both the slope and the y-intercept of the line.- **Slope (m):** This is the steepness of the line, showing how much y changes for a change in x. A positive slope means the line rises, while a negative slope means it falls. - **Y-intercept (b):** This is where the line crosses the y-axis. It tells you the value of y when x is 0.By rearranging any linear equation into this form, you can quickly identify these characteristics, making it easier to graph accurately and understand the relationship represented by the equation.

Consistency of Systems

A system of equations can be classified based on whether they produce a solution and if so, how many: - **Consistent Systems:** These have at least one solution. If two lines intersect at any single point, the system is consistent and has one solution. - **Inconsistent Systems:** These have no real solution. If lines are parallel and never intersect, the system is inconsistent. - **Dependent Systems:** These have an infinite number of solutions. This happens when the two lines are actually the same line, overlapping entirely on the graph. Understanding these terms helps determine the nature of the solutions when dealing with multiple equations.

Intersection of Lines

The intersection of lines represents the point where two lines cross each other on the graph. This point is crucial because it signifies the solution to the system of equations.

• If lines intersect at a single point, that point is the only solution, making the system consistent.
• Parallel lines never meet, indicating no solution and making the system inconsistent.
• Coinciding lines, or overlapping lines, share all points, indicating infinite solutions and a dependent system.

Finding the intersection through graphing allows us to visually verify solutions, reinforcing the conceptual understanding of algebraic results.