Problem 91 - Precalculus

Question

Apply a graphing utility to graph the two equations \(\frac{1}{3} x-\frac{5}{12} y=\frac{5}{6}\) and \(\frac{3}{7} x+\frac{1}{14} y=\frac{29}{28}\). Approximate the solution to this system of linear equations.

Step-by-Step Solution

Answer

The approximate solution is \((5.29, 0.12)\).

Step 1: Convert equations to slope-intercept form

Start by manipulating each equation to express them as functions of the form \(y = mx + b\). For the first equation, \(\frac{1}{3}x - \frac{5}{12}y = \frac{5}{6}\), solving for \(y\) gives us \(y = \frac{2}{5}x - 2\). For the second equation, \(\frac{3}{7}x + \frac{1}{14}y = \frac{29}{28}\), solving for \(y\) gives us \(y = -6x + 58\).

Step 2: Graph the equations

Using a graphing utility like a graphing calculator or online graph tool, enter the equations \(y = \frac{2}{5}x - 2\) and \(y = -6x + 58\). Plot both lines on the same graph.

Step 3: Identify point of intersection

Look for the point on the graph where the two lines intersect. This point is the solution to the system of equations as it satisfies both equations simultaneously.

Step 4: Estimate the intersection point

Determine the approximate coordinates of the intersection point by observing the graph. This will provide the approximate solution to the system. The intersection point is near \((5.29, 0.12)\).

Key Concepts

Graphing Utilities

Graphing utilities are powerful digital tools that help visualize mathematical equations and functions. They make solving problems involving systems of linear equations much simpler by providing a visual representation of each equation.
These tools include graphing calculators and software like Desmos or GeoGebra, which allow users to input equations and observe their graphs on a coordinate plane.
By graphing the equations, it becomes easier to see where they intersect, offering a precise method to find the solution. Benefits of using graphing utilities include:

• Quick visualization of equations to gauge solution possibilities.
• Ease of manipulation, where you can zoom in or out to better see details.
• Ability to handle complex equations that might be cumbersome to graph by hand.

Slope-Intercept Form

The slope-intercept form is a way to express a linear equation as: \[ y = mx + b \]where \(m\) represents the slope and \(b\) is the y-intercept.
This format is particularly useful for graphing because it readily provides the slope, which indicates the steepness of the line, and the y-intercept, where the line crosses the y-axis.When solving systems of equations, converting each equation to the slope-intercept form aids in plotting them on a graph efficiently.
For example, the given equations were transformed into:

• \(y = \frac{2}{5}x - 2\)
• \(y = -6x + 58\)

These forms reveal that the first line has a gentle slope, while the second line is much steeper.

Solutions to Equations

In the context of linear equations, a solution is a set of values that satisfies all the equations in the system simultaneously. For a system with two equations, a single solution will typically be a point where both lines intersect on a graph.
This means that the x and y coordinates of this intersection are the values that make both equations true at the same time.Using our earlier graphing example:

• The graph showed that the lines intersect near \((5.29, 0.12)\).
• These coordinates are the solution to the system, verifying that both equations hold true when these values are substituted in.

Identifying the solution on a graph gives a visible confirmation of where both linear patterns meet.

Intersection Point

The intersection point in a system of linear equations is the coordinate where both lines meet, marking the solution for the system. This point represents where both equations equally share the same values for x and y.
Finding this point is crucial as it unveils the answer to the question of where two systems of equations agree.In our case:

• The equations we graphed crossed paths at approximately \((5.29, 0.12)\).
• This intersection point is where the results from both equations overlap, making it the exact solution we were seeking.

Using graphing utilities greatly facilitates locating this point without relying solely on algebraic manipulation.