Problem 43 - College Algebra

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. $$ \begin{aligned} x+2 y &=7 \\ 2 x+6 y &=12 \end{aligned} $$

Step-by-Step Solution

Answer

The system is consistent with one solution.

Step 1: Understand the Equations

We have two equations: \(x + 2y = 7\) and \(2x + 6y = 12\). Our goal is to graph these equations and determine the relationship between them.

Step 2: Convert to Slope-Intercept Form

We convert each equation to the slope-intercept form \(y = mx + b\). For \(x + 2y = 7\), we solve for \(y\): \[ 2y = -x + 7 \] \[ y = -\frac{1}{2}x + \frac{7}{2} \] Next, for \(2x + 6y = 12\), we solve for \(y\): \[ 6y = -2x + 12 \] \[ y = -\frac{1}{3}x + 2 \].

Step 3: Graph the Equations

On a coordinate plane, graph the line \(y = -\frac{1}{2}x + \frac{7}{2}\). This line has a y-intercept at \(\frac{7}{2}\) and a slope of \(-\frac{1}{2}\). Next, graph the line \(y = -\frac{1}{3}x + 2\) with a y-intercept at \(2\) and a slope of \(-\frac{1}{3}\). Observe how the lines are positioned with respect to each other.

Step 4: Analyze Line Relationship

The lines have different slopes (\(-\frac{1}{2}\) and \(-\frac{1}{3}\)), indicating they are not parallel and will intersect at exactly one point. Thus, the system is consistent with one solution.

Key Concepts

Consistent System

A consistent system of equations is one where at least one solution exists. When graphing, if the lines intersect at any point, the system is considered consistent. This means that the lines are not parallel; instead, they meet at a specific point in the coordinate plane. For our exercise, upon converting the equations to slope-intercept form and graphing them, we observed that the lines intersect. Therefore, the system is consistent. In mathematical problems involving systems of equations, seeing this intersection is crucial as it immediately tells us that the equations work together and share a common solution. This shared solution is where both equations hold true simultaneously.

Slope-Intercept Form

The slope-intercept form is a way of writing linear equations as:

• \( y = mx + b \)

Here, \( m \) represents the slope of the line, and \( b \) indicates the y-intercept, which is where the line crosses the y-axis. Conversion to this form is helpful because it makes graphing easier, as you can directly see the slope and y-intercept from the equation.
In our exercise, the equations \(x + 2y = 7\) and \(2x + 6y = 12\) were converted into \(y = -\frac{1}{2}x + \frac{7}{2}\) and \(y = -\frac{1}{3}x + 2\), respectively.
These transformations tell us both how steep the lines are and where they cross the y-axis. This conversion is a fundamental step in solving systems of equations through graphing.

Intersection of Lines

The intersection of lines is an important concept in systems of equations. It is the point where two lines cross each other on a graph. These points of intersection represent solutions to the system of equations, meaning they satisfy both equations simultaneously.
In our exercise, once the lines \(y = -\frac{1}{2}x + \frac{7}{2}\) and \(y = -\frac{1}{3}x + 2\) were graphed, we observed that they intersect at a single point. This intersection confirms that a solution exists for the system of equations.

• The different slopes of \(-\frac{1}{2}\) and \(-\frac{1}{3}\) ensure that the lines will cross each other, indicating that they are not parallel.
• Thus, they meet in one unique location on the graph, showing precisely where both equations have a common solution.

Unique Solution

A unique solution occurs when there is exactly one set of values for the variables that satisfies all equations in the system. In the context of graphing, a unique solution is where the lines intersect at precisely one point.
In our example, the sloped lines with equations \(y = -\frac{1}{2}x + \frac{7}{2}\) and \(y = -\frac{1}{3}x + 2\) intersect only once. This tells us the system has a unique solution.

• The presence of distinct slopes assures us that each line proceeds in a slightly different direction and therefore must meet at some point.
• This meeting point, the intersection, is the unique solution and is vital for solving these equations.

Finding a unique solution means understanding that for a single pair of \(x\) and \(y\), both equations are satisfied, and outside this point, the equations do not concurrently hold true.