Problem 36 - Intermediate Algebra

Question

Use your graphing calculator to help determine the solution set for each of the following systems. Be sure to check your answers. (a) \(\left(\begin{array}{l}3 x-y=30 \\ 5 x-y=46\end{array}\right)\) (b) \(\left(\begin{array}{l}1.2 x+3.4 y=25.4 \\ 3.7 x-2.3 y=14.4\end{array}\right)\) (c) \(\left(\begin{array}{l}1.98 x+2.49 y=13.92 \\ 1.19 x+3.45 y=16.18\end{array}\right)\) (d) \(\left(\begin{array}{l}2 x-3 y=10 \\ 3 x+5 y=53\end{array}\right)\) (e) \(\left(\begin{array}{l}4 x-7 y=-49 \\ 6 x+9 y=219\end{array}\right)\) (f) \(\left(\begin{array}{l}3.7 x-2.9 y=-14.3 \\ 1.6 x+4.7 y=-30\end{array}\right)\)

Step-by-Step Solution

Answer

For each system, graph the equations and find their intersection.

Step 1: Input the equations into the graphing calculator

Enter the given equations into your graphing calculator. For example, for part (a), enter the equations \( y = 3x - 30 \) and \( y = 5x - 46 \) into the calculator, as these represent the two lines you will graph to find their intersection.

Step 2: Graph the equations

Graph the equations on your calculator. You should see each equation represented as a line. The solution to the system of equations is the point where the two lines intersect.

Step 3: Identify the point of intersection

Use the 'Intersect' function (often found under the 'Calc' menu) on your graphing calculator to determine the precise point where the two lines intersect. This is your solution.

Step 4: Verify the solution algebraically

Take the coordinates of the intersection point and substitute them back into the original equations to ensure they satisfy both equations. If they do, then your solution is correct.

Key Concepts

Graphing Calculator

A graphing calculator is an essential tool for students solving systems of equations. It helps visualize the equations by plotting them as lines on a coordinate plane. By entering the equations into the calculator, students can see how the lines behave and where they intersect, which is crucial for finding solutions to the system.

To input equations into the graphing calculator, first rearrange them in the form of \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. For example, take the equation from part (a): \( 3x - y = 30 \). Rearrange it to \( y = 3x - 30 \) for input into the calculator. Repeat for the second equation \( 5x - y = 46 \) to \( y = 5x - 46 \).

Once the equations are entered in this form, graphing is straightforward. Simply use the graphing function of the calculator to view the plotted lines, making sure your window settings allow you to see a good section of the plane where the lines intersect.

Intersection Point

The intersection point of two lines on a graph represents the solution to the system of equations. It is the specific set of coordinates \( (x, y) \) that satisfies both equations simultaneously. Identifying this point when graphing is a practical way to solve systems, especially when equations are complex.

Once the lines are plotted using the graphing calculator, locating the intersection point involves using the 'Intersect' feature. Typically found under the calculator’s 'Calc' menu, this function automatically calculates the exact coordinates where the lines cross. Make sure your calculator's window view is adjusted properly to show the point where they meet.

When the graphing calculator displays the intersection point, write down these coordinates. They are critical as they effectively represent the solution to your system of equations.

Algebraic Verification

Even after graphing and finding the intersection point, it's crucial to verify that this point is indeed the solution by using algebraic methods. This step ensures the accuracy of your solution and helps reinforce your understanding of the relationship between the equations.

To verify: take the intersection point obtained from the calculator, normally in the form of \( (x, y) \), and substitute these values back into the original equations. For example, if the point is \( (4, 2) \), substitute \( x = 4 \) and \( y = 2 \) into each equation from the system.

Check each one separately:

• Solve the first equation by substituting to see if both sides equal after simplification.
• Next, substitute the same \( x \) and \( y \) into the second equation. It should also hold true.

If both equations are satisfied with the substituted values, you have correctly identified the solution. This step of algebraic verification confirms accuracy and builds confidence in using multiple techniques to solve systems of equations.