Problem 58 - Intermediate Algebra

Question

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.) $$ \left\\{\begin{array}{l} y=-2 \\ y=\frac{2}{3} x-\frac{4}{3} \end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system is the point \((-1, -2)\). The system is consistent and independent.

Step 1: Understand the Equations

The system consists of two equations: 1. \( y = -2 \) is a horizontal line where the y-value is -2 for all x-values.2. \( y = \frac{2}{3}x - \frac{4}{3} \) is a linear equation in slope-intercept form with a slope of \( \frac{2}{3} \) and a y-intercept of \( -\frac{4}{3} \).

Step 2: Graph the Equations

On a coordinate plane, plot the line \( y = -2 \) as a horizontal line passing through points (0, -2) and (1, -2).Next, plot the line \( y = \frac{2}{3}x - \frac{4}{3} \). Start at point (0, -\frac{4}{3}) for the y-intercept, then use the slope to find another point. From (0, -\frac{4}{3}), rise 2 units and run 3 units to the right to plot another point at (3, 0). Connect these points with a straight line.

Step 3: Identify the Intersection

The intersection point of the two lines on the graph is where the solution to the system occurs.From the graphing done in Step 2, observe where the lines cross. The intersection has the coordinates \((-1, -2)\) from the graph.

Step 4: Check for Consistency and Dependency

Since the lines intersect at exactly one point, the system is consistent and independent. This means there is a unique solution to the system.

Key Concepts

Linear Systems

A linear system consists of two or more linear equations that use the same set of variables. Solving a linear system means finding all variable values that make each equation true simultaneously. In our exercise, the system is comprised of two equations:

• Equation 1: \( y = -2 \), a horizontal line indicating that the y-value is always -2, regardless of the x-value.
• Equation 2: \( y = \frac{2}{3}x - \frac{4}{3} \), a line with a slope and an intercept.

Linear systems can be solved using different methods such as graphing, substitution, or elimination. When using graphing, the solutions are points where the graphs intersect, revealing the values for variables like x and y. It's important to note that the lines in a linear system can:

• Intersect at one point, indicating a unique solution.
• Be parallel, indicating no solution.
• Overlap completely, indicating infinitely many solutions.

Slope-Intercept Form

The slope-intercept form of a linear equation is expressed as \( y = mx + b \). It's a simple and powerful way to describe a line algebraically. Here, \( m \) represents the slope of the line and \( b \) is the y-intercept.Understanding the slope:

• The slope, \( m \), tells you how steep the line is.
• It indicates the change in y for a one-unit change in x.
• A positive slope means the line rises to the right, while a negative slope means it falls to the right.

The y-intercept, \( b \), specifies where the line crosses the y-axis. In our exercise, for the equation \( y = \frac{2}{3}x - \frac{4}{3} \), the slope is \( \frac{2}{3} \) and the y-intercept is \( -\frac{4}{3} \). This helps us easily plot the line on a graph: start at (0, -\( \frac{4}{3} \)) and use the slope to find another point.

Inconsistent and Dependent Systems

When dealing with linear systems, it's critical to determine whether they are consistent or inconsistent, or if they are independent or dependent.Consistent vs. Inconsistent Systems:

• A consistent system has at least one solution. The lines intersect at some point.
• An inconsistent system has no solutions. The lines are parallel and never meet.

Independent vs. Dependent Systems:

• An independent system has exactly one solution. In graph terms, the lines meet at a single point.
• A dependent system has infinitely many solutions. It occurs when the lines overlap completely, being essentially the same line.

From our exercise, the system with equations \( y = -2 \) and \( y = \frac{2}{3}x - \frac{4}{3} \) is consistent and independent because they intersect at one unique point, giving a single solution.