Problem 53 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the two equations. Use the graphs to approximate the solution of the system. Round your results to three decimal places. $$\left\\{\begin{array}{r} 6 y=42 \\ 6 x-y=16 \end{array}\right.$$

Step-by-Step Solution

Answer

The graphs of the two equations intersect at \((x, y) = (3.833, 7)\), which is the solution of the system, rounded to three decimal places.

Step 1: Rewrite the equations in slope-intercept form

The first step is to convert the equations \(6y = 42\) and \(6x - y = 16\) to the slope-intercept form, \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. This enables us to get a clear visual representation when we graph the equations. For the first equation, rewrite as \(y = 7\). For the second equation, rewrite as \(y = 6x - 16\).

Step 2: Graph the equations

Plot the two equations obtained in the first step on the graph. For \(y = 7\), this line is a horizontal line at y = 7. For \(y = 6x - 16\), this line has a slope of 6 and crosses the y-axis at -16.

Step 3: Approximate the solution

Identify the point where the two lines intersect, which is the solution to the system of equations. Use your graphing utility to get a numeric exponential of this intersection point, rounding to three decimal places.

Key Concepts

Graphing

When working with systems of equations, graphing can provide a visual representation, making it easier to solve.
To start, we first need to plot each equation on a coordinate plane. Once plotted, the graphical intersection of these lines indicates the solution to the system.
The graphing method not only shows where the lines meet but also helps in verifying the accuracy of other algebraic solutions.

• For a line represented by an equation, you need to identify key features like the slope and y-intercept.
• Use these features to draw the line accurately on your graph.

Once the lines intersect, this point provides the values of x and y that satisfy both equations.

Slope-intercept form

The slope-intercept form is a popular way to express linear equations. It is represented as:
\( y = mx + c \)
Here, \( m \) is the slope of the line, and \( c \) is the y-intercept, the point where the line crosses the y-axis. This format makes it straightforward to graph linear equations.
For example, consider the equations given:

• The first equation, \( 6y = 42 \), simplifies to \( y = 7 \), which is a horizontal line on the graph.
• The second equation, \( 6x - y = 16 \), reformulates to \( y = 6x - 16 \), providing a slope of 6 and a y-intercept of -16.

Knowing how to convert equations into this form helps analyze and solve systems of equations efficiently.

Intersection point

Finding the intersection point of two lines is crucial in solving systems of equations graphically.
This point represents the values of \( x \) and \( y \) that satisfy both equations simultaneously.
For instance, when the graph of the equation \( y = 7 \) intersects with \( y = 6x - 16 \), the intersection gives the required solution.

• The intersection is calculated where both equations' lines meet on the graph.
• Using a graphing calculator or plotting tool can provide a more accurate intersection point.

The coordinates at this intersection are typically written as an ordered pair \( (x, y) \), representing the system's solution.