Problem 28 - Intermediate Algebra

Question

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. See Examples 3 and 4. $$ \left\\{\begin{array}{l} 3 x=5-2 y \\ 3 x+2 y=7 \end{array}\right. $$

Step-by-Step Solution

Answer

The system is inconsistent; the lines are parallel and do not intersect.

Step 1: Write Each Equation in Slope-Intercept Form

Start by rearranging each equation to the slope-intercept form, which is \( y = mx + b \). This makes it easy to graph the equations.\\- For the first equation: \[ 3x = 5 - 2y \] Rearranging gives: \[ 2y = 5 - 3x \] Divide both sides by 2: \[ y = -\frac{3}{2}x + \frac{5}{2} \]- For the second equation: \[ 3x + 2y = 7 \] Rearranging gives: \[ 2y = 7 - 3x \] Divide both sides by 2: \[ y = -\frac{3}{2}x + \frac{7}{2} \]

Step 2: Graph the Equations

Using the slope-intercept form of each equation, plot the y-intercept and then use the slope to find another point on the line. - For the equation \( y = -\frac{3}{2}x + \frac{5}{2} \), the y-intercept is at \( \frac{5}{2} \) and the slope is \(-\frac{3}{2}\). - For the equation \( y = -\frac{3}{2}x + \frac{7}{2} \), the y-intercept is at \( \frac{7}{2} \) and the slope is the same, \(-\frac{3}{2}\). When you plot both lines, observe their position.

Step 3: Analyze the Graph

Since after graphing, both lines appear to be parallel and do not intersect, this indicates something about the system of equations. Due to having the same slope but different y-intercepts, the lines never meet.

Step 4: Determine System Characteristics

The two lines are parallel and have no intersection point. This means the system has no solutions and is considered inconsistent.

Key Concepts

Graphing Method

Solving a system of equations by graphing is a visual approach to finding solutions. This method involves plotting each equation on a coordinate plane and identifying the point where the lines intersect. Here's how it works:

• First, you convert each equation into a format that is easy to graph.
• Next, plot the equations on the graph to find their points of intersection.
• The intersection point represents the solution to the system.

When the lines intersect at a point, the coordinates of this point give the solution to the system. If the lines are parallel or coincident, they indicate either no solution or infinitely many solutions, respectively.

Slope-Intercept Form

To make the graphing method simpler, we convert equations into the slope-intercept form, expressed as \(y = mx + b\). This form is intuitive because:

• \(m\) represents the slope of the line, showing how steep the line is.
• \(b\) is the y-intercept, where the line crosses the y-axis.

Once an equation is in this form, graphing becomes straightforward:

• Start plotting at the y-intercept on the graph.
• Use the slope to determine the next points on your line by moving up or down and left or right based on the slope's ratio.

For example, in our equations, we find that both lines have a slope of \(-\frac{3}{2}\), showing they rise down as they move right.

Inconsistent Systems

An inconsistent system arises when two lines never meet, implying there is no single solution that satisfies both equations. This typically results when the two lines are parallel, as they will not intersect at any point on the graph.

• If the lines have the same slope but different y-intercepts, they run parallel.
• Parallel lines on a graph indicate that the system is inconsistent.
• No intersection means no solution to the system of equations.

Recognizing an inconsistent system early can save time, as there is no need to search for a solution that doesn't exist.

Parallel Lines

Parallel lines are lines in a plane that never intersect. They always maintain the same distance apart and point in the same direction, which is typically determined by their slope:

• They have identical slopes but separate y-intercepts.
• This is a key indicator of parallelism in graphing.

In practice, when graphing two lines from a system of equations and their slopes turn out the same but they never cross, those lines are parallel. From this, we can deduce that the system of equations has no common solution, marking it as inconsistent. This visual confirmation is crucial for understanding the nature of the equations being solved.