Problem 46 - 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.) $$ \left\\{\begin{array}{l} y=-\frac{5}{6} x+2 \\ \frac{5}{2} x+3 y=6 \end{array}\right. $$

Step-by-Step Solution

Answer

The system is dependent with infinitely many solutions.

Step 1: Rewrite the Second Equation in Slope-Intercept Form

Start by rewriting the second equation \( \frac{5}{2} x + 3y = 6 \) in slope-intercept form (\( y = mx + b \)). Subtract \( \frac{5}{2}x \) from both sides to get \( 3y = -\frac{5}{2}x + 6 \). Then, divide every term by 3 to solve for \( y \):\[y = -\frac{5}{6}x + 2 \]

Step 2: Compare Equations for Dependency

Now, compare the equation from Step 1, \( y = -\frac{5}{6}x + 2 \), to the original first equation, \( y = -\frac{5}{6}x + 2 \). Both equations are identical.

Step 3: Interpret the Results

Since both equations are the same, every point that lies on the line described by these equations is a solution to the system. This means the system is dependent, and there are infinitely many solutions.

Key Concepts

Solving by Graphing

Solving systems of linear equations by graphing is a visually intuitive method. It involves plotting each equation on the same graph to find the point of intersection, which represents the solution. Here's how it works:

• First, transform each equation into slope-intercept form, which is easier to graph.
• The graph of an equation is a straight line. Where these lines intersect is the solution of the system.

This method provides a clear visual representation of solutions. If two equations result in parallel lines, they do not intersect, meaning there is no solution, and the system is inconsistent. If they lie on top of each other, the system is dependent.

Dependent System

A dependent system of equations occurs when all equations in the system actually describe the same line. In other words, one equation can be derived from the other or others by simple algebraic manipulation. This leads to infinitely many solutions because every point on the line is a solution.
To determine dependency, follow these steps:

• Rewrite equations to a comparable form, such as the slope-intercept form.
• Examine the slopes and intercepts. If they are identical, the system is dependent.

In a dependent system, rather than a single point of intersection, the solution is the entire line.

Slope-Intercept Form

The slope-intercept form of a linear equation is expressed as \( y = mx + b \), where \( m \) represents the slope and \( b \) is the y-intercept. This form is particularly useful for graphing because:

• The value \( m \) tells us the steepness or incline of the line. A positive slope rises from left to right, while a negative slope falls.
• The value \( b \) is the point where the line crosses the y-axis, providing a starting point for graphing.

By converting equations into this form, you can quickly and easily plot them on a graph. This form is integral in identifying if two lines are parallel (same slope), intersecting (different slopes), or coincident (same slope and y-intercept).