Problem 56 - Algebra A Combined Function

Question

Use a graphing calculator to solve each system. $$ \left\\{\begin{array}{l} x+y=-15.2 \\ -2 x+5 y=-19.3 \end{array}\right. $$

Step-by-Step Solution

Answer

Solve the system by graphing both equations and finding the intersection point.

Step 1: Graph the First Equation

First, solve the equation \( x + y = -15.2 \) for \( y \):\[ y = -x - 15.2 \]On the graphing calculator, enter this equation as \( y_1 = -x - 15.2 \). This sets up the line to be graphed.

Step 2: Graph the Second Equation

Next, solve the equation \( -2x + 5y = -19.3 \) for \( y \):First, isolate \( y \):\[ 5y = 2x -19.3 \]Then divide by 5:\[ y = \frac{2}{5}x - \frac{19.3}{5} \approx 0.4x - 3.86 \]Enter this equation into the graphing calculator as \( y_2 = 0.4x - 3.86 \). This will be the second line on the graph.

Step 3: Find the Intersection Point

Once both equations are entered, use the graphing calculator to plot both lines. The intersection point of these two lines provides the solution to the system of equations. Adjust the viewing window if necessary to clearly observe where the lines intersect. Utilize the calculator's 'intersect' function to determine the exact coordinates of the intersection point.

Step 4: Verify the Solution

Check the intersection point from the calculator. If the point is \((a, b)\), substitute back into the original equations:1. For \( x + y = -15.2 \): check if \( a + b = -15.2 \).2. For \(-2x + 5y = -19.3 \): check if \( -2a + 5b = -19.3 \).If both conditions hold, then \((a, b)\) is the correct solution.

Key Concepts

Graphing Calculator

A graphing calculator is a powerful tool that can help solve systems of linear equations visually and efficiently. These calculators can plot multiple equations on the same graph, enabling a clear representation of their relationship. To use a graphing calculator for graphing systems of equations, you need to:

• Input each equation separately, ensuring they are solved for the dependent variable (usually \(y\)).
• Graph the equations on the same coordinate plane. This allows you to visually assess where the solutions for the system may lie.
• Use the calculator's specific function to calculate the 'intersect' of the graphed lines. This function will determine where the equations meet, providing exact coordinates.

By using a graphing calculator, you not only find a solution quickly but also develop a stronger visual understanding of the relationships between equations. This approach helps reinforce the concept of solving algebraic systems through graphical representation.

Intersection Point

The intersection point of two lines on a graph represents the solution to a system of linear equations. This point is where both lines, each representing an equation, cross each other on the coordinate plane. Finding the intersection visually demonstrates how both equations satisfy the same set of conditions at this particular point.
To find the intersection point using a graphing calculator, plot each equation:

• Ensure the graphing window is correctly sized to view potential intersection points. Adjusting the window can sometimes reveal intersections that were initially out of view.
• Use the calculator's 'intersect' feature to calculate the exact coordinates. The calculator will generally return a pair of \((x, y)\) values that satisfy both equations.

This coordinate pair is important because it confirms that when these values are plugged into both original equations, they hold true, verifying the intersection point as a valid solution.

Solving Linear Equations

Solving linear equations involves finding values for variables that make each equation true. In systems of equations, these variables often signify where two or more equations intersect within a specific plane. Here’s how you solve such equations:

• Rearrange each equation into a form that makes it easier to graph, typically \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
• Implement substitution or elimination methods if you're not graphing to solve algebraically. However, graphing offers a visual solution through intersection points.
• Verify your solutions by substituting them back into the original equations to ensure they satisfy each equation. This step confirms the correctness of the solution.

Solving linear equations, especially through graphical methods, provides insight into the behavior of equations as they relate to one another on a coordinate plane. The visual solution through intersection highlights the tangible connection between algebraic concepts and graphical representation.