Problem 13 - Intermediate Algebra

Question

For Problems \(1-16\), use the graphing approach to determine whether the system is consistent, the system is inconsistent, or the equations are dependent. If the system is consistent, find the solution set from the graph and check it. (Objective 1) $$ \left(\begin{array}{l} y=2 x+5 \\ x+3 y=-6 \end{array}\right) $$

Step-by-Step Solution

Answer

The system is consistent and independent with the solution set being the intersection point from the graph.

Step 1: Rewrite Each Equation

Start by rewriting each equation in a simpler form if necessary. The first equation is already in slope-intercept form, \( y = 2x + 5 \). The second equation, \( x + 3y = -6 \), can be rewritten as \( y = -\frac{1}{3}x - 2 \) by solving for \( y \).

Step 2: Graph the First Equation

Plot the first equation \( y = 2x + 5 \) on a coordinate plane. The y-intercept is 5, so plot the point (0, 5). The slope is 2, meaning from this point move 1 unit to the right and 2 units up to plot the next point.

Step 3: Graph the Second Equation

Plot the second equation \( y = -\frac{1}{3}x - 2 \) on the same coordinate plane. The y-intercept is -2; plot the point (0, -2). With a slope of -1/3, move 3 units to the right and 1 unit down to find another point.

Step 4: Determine Intersection Point

Observe where the two lines intersect on the graph. This point of intersection is the solution to the system of equations since it satisfies both equations simultaneously.

Step 5: Analyze Intersection

The lines intersect at one point, indicating that the system is consistent and independent. Note the coordinates of the intersection point as the solution.

Step 6: Verify the Solution

Substitute the coordinates of the intersection point back into the original equations to ensure they are satisfied. This confirms that the solution is correct and both equations are solved.

Key Concepts

Consistent Systems

When dealing with a system of equations, a consistent system is one that has at least one solution. This means that the graphs of the equations will intersect at least at one point on the coordinate plane. For two linear equations, this solution is the point where both lines cross each other.

To identify a consistent system, through graphing, follow these steps:

• Graph each equation on the same coordinate plane.
• Look for the intersection point of these lines. This point represents the solution that satisfies both equations simultaneously.
• Verify this intersection by substituting its coordinates back into the original equations. If both equations hold true with these values, the solution is correct.

A key tip is that a consistent system can either be independent or dependent. An independent consistent system will have exactly one intersection point as its solution.

Inconsistent Systems

An inconsistent system is one that has no solution. This happens when two or more linear equations represent parallel lines on the graph. Parallel lines do not intersect, which visually shows us that there are no common values that satisfy both equations simultaneously.

Here's how to identify an inconsistent system:

• During graphing, you will see that the lines never meet or touch. This implies no shared solution.
• Algebraically, lines are parallel when their slopes are equal but their y-intercepts are different.

In practice, consistently verifying your graph calculations can help avoid misclassifying a system as inconsistent. It's important to note that real-world problems can have chances for cross-verification with other methods like substitution or elimination.

Dependent Equations

Dependent equations are a special case in systems of equations where the equations describe the same line. This means any solution that satisfies one equation will also satisfy the other, indicating that they have infinitely many solutions.

For spotting dependent equations:

• Graph both equations. If they overlay perfectly, you've encountered dependent equations.
• Check if one equation can be rewritten as a scalar multiple of the other; this often indicates they are dependent.

In practical work, picking different points on the line and checking them in both equations acts as a robust verification. Don't forget: dependent equations are consistent since they contain solutions infinitely at each overlapping point, unlike inconsistent systems which contain none.