Problem 93 - Algebra for College Students

Question

For each of the systems of equations in Problem 92, use your graphing calculator to help determine whether the system is consistent or inconsistent or whether the equations are dependent. (a) \(\left(\begin{array}{l}y=3 x-1 \\ y=9-2 x\end{array}\right)\) (b) \(\left(\begin{array}{rl}5 x+y & =-9 \\ 3 x-2 y & =5\end{array}\right)\) (c) \(\left(\begin{array}{l}4 x-3 y=18 \\ 5 x+6 y=3\end{array}\right)\) (d) \(\left(\begin{array}{l}2 x-y=20 \\ 7 x+y=79\end{array}\right)\) (e) \(\left(\begin{array}{l}13 x-12 y=37 \\ 15 x+13 y=-11\end{array}\right)\) (f) \(\left(\begin{array}{l}1.98 x+2.49 y=13.92 \\ 1.19 x+3.45 y=16.18\end{array}\right)\)

Step-by-Step Solution

Answer

All systems are consistent with unique solutions.

Step 1: Identifying System (a)

The given system of equations is:1. \( y = 3x - 1 \)2. \( y = 9 - 2x \)These are two linear equations in slope-intercept form \( y = mx + b \). We need to graph them to find their intersection.

Step 2: Graphing Equations (a)

Use a graphing calculator to plot the equations. - The first line has a slope of 3 and a y-intercept of -1. - The second line has a slope of -2 and a y-intercept of 9. Check the intersection point of these two lines.

Step 3: Conclusion for System (a)

The intersection point of the two lines occurs at \( x = 2, y = 5 \). The system is consistent as there is a unique solution at this point.

Step 4: Identifying System (b)

The given system of equations is:1. \( 5x + y = -9 \)2. \( 3x - 2y = 5 \)Convert these to slope-intercept form to prepare for graphing.

Step 5: Graphing Equations (b)

Rearrange both equations to solve for \( y \):- \( y = -5x - 9 \)- \( y = \frac{3}{2}x - \frac{5}{2} \)Graph these equations using the slope and y-intercept.

Step 6: Conclusion for System (b)

The lines intersect at a single point, indicating they are consistent with a unique solution.

Step 7: Identifying System (c)

The given system of equations is:1. \( 4x - 3y = 18 \)2. \( 5x + 6y = 3 \)Convert these to slope-intercept form.

Step 8: Graphing Equations (c)

Solve for \( y \):- \( y = \frac{4}{3}x - 6 \)- \( y = -\frac{5}{6}x + \frac{1}{2} \)Graph these equations.

Step 9: Conclusion for System (c)

The lines intersect at a single point, thus the system is consistent with a unique solution.

Step 10: Identifying System (d)

The given system of equations is:1. \( 2x - y = 20 \)2. \( 7x + y = 79 \)Convert into slope-intercept form for graphing.

Step 11: Graphing Equations (d)

Rewrite the equations:- \( y = 2x - 20 \)- \( y = -7x + 79 \)Graph both lines.

Step 12: Conclusion for System (d)

The lines cross at a single point, thus the system is consistent with a unique solution.

Step 13: Identifying System (e)

The given system of equations is:1. \( 13x - 12y = 37 \)2. \( 15x + 13y = -11 \)We will convert to slope-intercept form to graph.

Step 14: Graphing Equations (e)

Solve for \( y \):- \( y = \frac{13}{12}x - \frac{37}{12} \)- \( y = -\frac{15}{13}x - \frac{11}{13} \)Graph the equations.

Step 15: Conclusion for System (e)

The lines intersect at one point, indicating that the system is consistent with a unique solution.

Step 16: Identifying System (f)

The given system of equations is:1. \( 1.98x + 2.49y = 13.92 \)2. \( 1.19x + 3.45y = 16.18 \)We will graph these to check for consistency.

Step 17: Graphing Equations (f)

Convert to slope-intercept form:- \( y = -\frac{1.98}{2.49}x + \frac{13.92}{2.49} \)- \( y = -\frac{1.19}{3.45}x + \frac{16.18}{3.45} \)Use a graphing calculator to plot and observe any intersections.

Step 18: Conclusion for System (f)

The lines intersect at a particular point, meaning the system is consistent with a unique solution.

Key Concepts

graphing calculator

A graphing calculator is a powerful tool for visualizing mathematical equations and analyzing their properties. When working with systems of equations, such as those provided in the exercise, a graphing calculator helps to plot each equation as a line on a coordinate plane. By examining where these lines cross, we can determine whether a system is consistent (the lines intersect), inconsistent (the lines do not intersect), or if the equations are dependent (the lines overlap entirely, representing the same equation).

Here's how you can use a graphing calculator effectively:

• Input each equation into the calculator, ensuring it is in slope-intercept form (\( y = mx + b \)).
• Graph the equations on the coordinate plane by selecting the graphing function.
• Observe where the lines intersect to identify solutions (if present).

Graphing calculators not only provide a visual confirmation of the equations' relationships but also enable precise calculations of intersection points, assisting in verifying the calculations you make by hand.

consistent systems

A consistent system of equations refers to a system that has at least one solution. In most cases, this means that the graphs of the equations intersect at a single point, which indicates a unique solution for the system. However, it is also possible for the system to have infinitely many solutions, which occurs when the equations are dependent.

To identify a consistent system using graphing or algebraic methods, you can:

• Find the slope-intercept form of each equation to facilitate graphing.
• Plot the equations and observe if they cross at any point.
• Check the intersection point, if any, to determine the specific (x,y) solution.

In the examples from the exercise, all systems were identified as consistent because their corresponding lines intersected at a specific point, indicating that there was a unique solution for each.

dependent equations

Dependent equations form a system where all equations describe the same line. As a result, these equations have infinitely many solutions, represented by any point on the line. To determine if equations are dependent, you should compare the coefficients:

• If the equations are multiples of each other, the lines overlap, showing they are dependent.
• Graph both equations to visually confirm overlapping lines.
• Alternatively, simplify the equations to find equivalent expressions, indicating dependency.

In the exercise provided, there were no dependent equations, but understanding this concept is crucial as it shows a unique relationship between equations that are not apparent when purely manipulating algebraic expressions.

slope-intercept form

The slope-intercept form of a linear equation is particularly useful when dealing with systems of equations because it allows for easy graphing and clear interpretations of equations. It is written as:\[y = mx + b\]where \(m\) represents the slope of the line, and \(b\) indicates the y-intercept, or the point where the line crosses the y-axis.

To convert any linear equation into this form:

• Solve for \(y\) in terms of \(x\).
• Ensure the coefficient of \(y\) is 1 by dividing the entire equation by the coefficient if necessary.
• Rearrange terms to isolate \(y\) on one side of the equation.

This form simplifies the process of graphing equations and finding intersection points. It also helps in understanding the relationships between different lines, such as parallel or perpendicular lines, based on their slopes.