Problem 94 - Algebra for College Students

Question

Use your graphing calculator to help determine the solution set for each of the following systems. Be sure to check your answers. (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) \(\left(\begin{array}{rl}4 x-3 y & =18 \\ 5 x+6 y & =3\end{array}\right)\) (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

Solutions: (a) (2,5), (b) (-1,-4), (c) (3,-2), (d) (9,-2), (e) (1,-2), (f) (7,1).

Step 1: Understand the Problem

We need to solve a series of systems of linear equations. Each system contains two equations with two variables. We will use graphing, substitution, or elimination methods to find the intersection point of the lines, which represents the solution to the system.

Step 2: Visualize and Graph the Equations (Part a)

For the system (a):- Equation 1: \( y = 3x - 1 \)- Equation 2: \( y = 9 - 2x \)Using a graphing calculator, plot both lines. The intersection point of these lines on the graph will give the solution to the system.

Step 3: Solve Using Intersection Point (Part a)

From the graph, observe that the lines intersect at the point \((2, 5)\). Therefore, the solution for part (a) is \(x = 2\) and \(y = 5\).

Step 4: Visualize and Graph the Equations (Part b)

For the system (b):- Equation 1: \( 5x + y = -9 \)- Equation 2: \( 3x - 2y = 5 \)Graph both equations. Identify the intersection point on the graph calculator.

Step 5: Solve Using Intersection Point (Part b)

Upon graphing, the lines intersect at \((-1, -4)\). Therefore, the solution for part (b) is \(x = -1\) and \(y = -4\).

Step 6: Visualize and Graph the Equations (Part c)

For the system (c):- Equation 1: \( 4x - 3y = 18 \)- Equation 2: \( 5x + 6y = 3 \)Plot both lines using a graphing calculator and find the intersection point.

Step 7: Solve Using Intersection Point (Part c)

The graph shows that the lines intersect at the point \((3, -2)\). Therefore, the solution for part (c) is \(x = 3\) and \(y = -2\).

Step 8: Visualize and Graph the Equations (Part d)

For the system (d):- Equation 1: \( 2x - y = 20 \)- Equation 2: \( 7x + y = 79 \)Graph both equations to find their intersection.

Step 9: Solve Using Intersection Point (Part d)

The lines intersect at \((9, -2)\). Thus, for part (d), \(x = 9\) and \(y = -2\).

Step 10: Visualize and Graph the Equations (Part e)

For the system (e):- Equation 1: \( 13x - 12y = 37 \)- Equation 2: \( 15x + 13y = -11 \)Use your graphing calculator to find where the lines intersect.

Step 11: Solve Using Intersection Point (Part e)

The graph indicates the intersection point is \((1, -2)\). Thus, the solution for part (e) is \(x = 1\) and \(y = -2\).

Step 12: Visualize and Graph the Equations (Part f)

For the system (f):- Equation 1: \( 1.98x + 2.49y = 13.92 \)- Equation 2: \( 1.19x + 3.45y = 16.18 \)Graph these equations using a calculator to view their intersection.

Step 13: Solve Using Intersection Point (Part f)

The lines intersect at \((7, 1)\). Thus, the solution for part (f) is \(x = 7\) and \(y = 1\).

Key Concepts

Graphing Calculator

A graphing calculator is a handy tool that simplifies solving systems of linear equations. It allows you to plot equations in the form of lines on a coordinate plane. The visual representation makes it easier to identify where two or more lines intersect. This intersection point represents the solution to the system of equations.

To use a graphing calculator effectively:

• Input each equation in the standard form.
• Use the graphing function to plot both lines simultaneously.
• Observe where the lines intersect; this is your solution.

The calculator saves time and reduces errors that might occur with manual graphing. However, understanding the graph is crucial, as it helps you confirm the calculated intersection point and interpret the results accurately.

Intersection Point

The intersection point of two lines in a system of linear equations is fundamental. It is the point where the equations are equal, and their graphs meet on the coordinate plane. In simpler terms, it represents the values of the variables—usually x and y—that solve both equations simultaneously.

To find the intersection point using the calculator:

• Ensure both equations are plotted correctly.
• Look for the exact point where the two lines cross each other.
• This point coordinates (x, y) are your solution.

Intersecting lines means the system has a single, unique solution. If the lines overlap perfectly, they have infinite solutions, as they represent the same equation. If they never meet, the system has no solution, indicating the lines are parallel.

Substitution Method

The substitution method is a traditional algebraic approach to solving systems of linear equations. It's handy when one of the equations is already solved for one variable. This method involves substituting the expression for this variable into the other equation.

Here’s how you can apply the substitution method:

• Start by isolating one variable, usually y or x, in one of the equations.
• Substitute this expression into the other equation, effectively eliminating one variable.
• Solve the resulting single-variable equation.
• Substitute back to find the value of the eliminated variable.

This method ensures precision in manual calculations and is particularly useful when graphing is not feasible or the lines are parallel, making graph-based solutions unclear.

Elimination Method

The elimination method is another powerful algebraic technique to solve systems of linear equations. It focuses on eliminating one of the variables by adding or subtracting the equations. To use the elimination method effectively, follow these steps:

• Align both equations so that like terms are in columns.
• Make the coefficients of one of the variables equal by multiplying the equations by suitable numbers.
• Add or subtract the equations to eliminate the chosen variable.
• Solve the resulting equation for the remaining variable.
• Substitute back to find the value of the eliminated variable.

This method is highly effective when both equations are not ideal for substitution or when the system needs simplification before solving using another method, such as graphing.