Problem 36 - Elementary Algebra Within Reach

Question

In Exercises 27-36, solve the system by graphing. $$ \left\\{\begin{array}{r} -x+\frac{2}{3} y=5 \\ 9 x-6 y=6 \end{array}\right. $$

Step-by-Step Solution

Answer

The system of equations has no solution as the lines on the graph of these two equations do not intersect.

Step 1: Convert Both Equations into slope-intercept form

The first equation can be rewritten as \(y = \frac{3}{2}x + 5\). The second equation can be rewritten as \(y = \frac{3}{2}x + 1\)

Step 2: Graph Both Equations

Now the equations are in slope-intercept form, draw them onto a coordinate plane. This will result in two lines. The line for the first equation will cross the y-axis at (0,5), and the line for the second equation will cross the y-axis at (0,1). The gradient or slope, represented by m in the equation y = mx + b, is the same for both lines (3/2), therefore the lines have the same direction or tilt.

Step 3: Find the Intersection Point of the Two Lines

In this case, the two lines on the graph will not intersect as they have the same slope but different y-intercepts. Therefore, the system of equations has no solution.

Key Concepts

Graphing Method

The graphing method is a visual approach to solving systems of equations. It involves drawing each equation as a line on a coordinate plane and identifying where the lines intersect. This intersection point, if it exists, represents the solution to the system, as it satisfies both equations simultaneously.
Graphing provides a clear visual representation of the relationship between the equations. However, this method can be less precise for finding exact solutions, especially when dealing with non-whole numbers or complex intersections. Even so, it elegantly shows when lines are parallel, intersecting, or overlapping.
The method simplifies the task of solving by showing the solutions graphically and helps in understanding the behavior and relationship of linear equations in a system.

Slope-Intercept Form

The slope-intercept form of a linear equation is a way of expressing the equation such that it is easy to graph. It is given by the formula: \[ y = mx + b \] where:

• \( m \) is the slope of the line, which represents the steepness and direction of the line.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

By rewriting equations into this form, you can quickly identify how the line behaves on a graph.
For example, in the original problem, converting the equations to \( y = \frac{3}{2}x + 5 \) and \( y = \frac{3}{2}x + 1 \) makes it straightforward to draw their graphs and understand their properties like parallelism and y-intercepts.

Parallel Lines

Parallel lines are lines in a plane that never meet. They have the same slope but different y-intercepts. This means they tilt in the same direction and share the same angle of inclination.
In a system of equations, if the lines are parallel, then they do not intersect. In the context of the graph, you will notice that no single point lies on both lines at the same time. That's why such systems are called inconsistent - they have no solutions.
In the given step-by-step solution, both equations, \( y = \frac{3}{2}x + 5 \) and \( y = \frac{3}{2}x + 1 \), have a slope of \( \frac{3}{2} \). Thus, they are parallel and will never intersect.

System of Linear Equations with No Solution

A system of linear equations has no solution when there are no points that satisfy all the equations simultaneously. This scenario typically occurs when the lines represented by the equations are parallel.
Such a system is described as 'inconsistent' because there is no common solution to both equations. This is visually recognizable using the graphing method, as you'll see the lines running parallel to each other, with no intersection point.
In our exercise, the system is composed of parallel lines because both equations share the same slope but different y-intercepts. Since parallel lines do not meet, this confirms that there is no solution to the system.