Problem 89 - Precalculus

Question

Apply a graphing utility to graph the two equations \(23 x+15 y=7\) and \(46 x+30 y=14 .\) Approximate the solution to this system of linear equations.

Step-by-Step Solution

Answer

The system has infinitely many solutions, as both equations represent the same line.

Step 1: Identify the Equations

We are given two linear equations: \(23x + 15y = 7\) and \(46x + 30y = 14\). We need to graph these equations using a graphing utility to find their intersection point.

Step 2: Convert to Slope-Intercept Form

Convert each equation to the form \(y = mx + b\). For \(23x + 15y = 7\), rearrange to get \(15y = -23x + 7\) then \(y = -\frac{23}{15}x + \frac{7}{15}\). For \(46x + 30y = 14\), rearrange to get \(30y = -46x + 14\) then \(y = -\frac{46}{30}x + \frac{14}{30}\). Simplify the second equation to \(y = -\frac{23}{15}x + \frac{7}{15}\).

Step 3: Input Equations into Graphing Utility

Use a graphing calculator or software to input both equations: \(y = -\frac{23}{15}x + \frac{7}{15}\) for both equations. Observe the graph and check where the lines intersect.

Step 4: Analyze the Graph

Both equations represent the same line since they have identical slope-intercept forms. Therefore, the system has infinitely many solutions as each point on the line is a solution to both equations.

Key Concepts

Graphing Utility

A graphing utility is a software tool that allows you to visualize mathematical equations by plotting them on a graph. It can be a graphing calculator or a computer program like Desmos or GeoGebra. When plotting equations, the utility can help identify key features such as intercepts and points of intersection with other graphs.

For this exercise, we used a graphing utility to determine where the lines represented by the equations intersect. By entering the slope-intercept form of each equation into the utility, we can visualize them:

• Both equations are entered as: \(y = -\frac{23}{15}x + \frac{7}{15}\)
• The utility will plot these graphs, showing their shapes and intersections clearly.
• It becomes evident that both equations actually graph the same line.

Using a graphing utility is an efficient way to solve and understand systems of equations as it visually reinforces the concept of intersection being a solution.

Slope-Intercept Form

The slope-intercept form of a linear equation is expressed as \(y = mx + b\), where \(m\) stands for the slope and \(b\) for the y-intercept. This form is particularly advantageous for graphing since it clearly shows how the line behaves.In our exercise, both equations were converted into the slope-intercept form:

• The equation \(23x + 15y = 7\) becomes \(y = -\frac{23}{15}x + \frac{7}{15}\)
• Similarly, the equation \(46x + 30y = 14\) simplifies to the same form: \(y = -\frac{23}{15}x + \frac{7}{15}\)

Notice how both lines share identical values for both the slope \(-\frac{23}{15}\) and y-intercept \(\frac{7}{15}\).

This similarity means they represent the same line. The slope-intercept form, therefore, makes it clear whether two lines are parallel or actually the same line, as in this case.

Infinitely Many Solutions

A system of equations can have one solution, no solution, or infinitely many solutions. When both equations represent the same line, as in this exercise, there are infinitely many solutions.

Understanding why there are infinitely many solutions involves recognizing that every point on the line satisfies both equations simultaneously:

• Since both equations simplify to \(y = -\frac{23}{15}x + \frac{7}{15}\), they describe the same set of points on a plane.
• Each point \((x, y)\) that lies on this line solves both equations because substituting into either equation gives the same result.
• The visual representation on the graph is just one line, without any intersections—that entire line is the solution.

This concept reinforces the fact that even though we have two equations, they are not distinct in their geometric representation, leading to infinite solutions since they coincide perfectly.