Problem 41 - Algebra and Trigonometry

Question

Use the graphical method to find all solutions of the system of equations, correct to two decimal places. \(\left\\{\begin{array}{l}{x^{2}+y^{2}=25} \\ {x+3 y=2}\end{array}\right.\)

Step-by-Step Solution

Answer

Solutions are approximately (3.94, -0.65) and (-4.37, 2.12).

Step 1: Understanding the System of Equations

We are given two equations: \(x^2 + y^2 = 25\) which represents a circle with a radius of 5 centered at the origin, and \(x + 3y = 2\) which is a linear equation. The task requires finding the intersection points of these two graphs, which represent the solutions to the system of equations.

Step 2: Graphing the Circle Equation

Draw the graph of the circle \(x^2 + y^2 = 25\), which is centered at the origin (0,0) with a radius of 5. This is a complete circle passing through points like (5,0), (0,5), (-5,0), and (0,-5).

Step 3: Graphing the Linear Equation

Sketch the line represented by \(x + 3y = 2\). To do this, find two points on the line: when \(x = 0\), \(y = \frac{2}{3}\), and when \(y = 0\), \(x = 2\). Draw the line passing through these points.

Step 4: Finding Intersection Points

The intersection points can be found by checking where the line intersects the circle. Solve the system graphically or analytically to find where the two curves overlap. Solving algebraically results in: Substitute \(y = \frac{2-x}{3}\) from the line equation into the circle equation, leading to: \(x^2 + (\frac{2-x}{3})^2 = 25\). Simplify and solve this quadratic equation to find values of \(x\).

Step 5: Calculating Precise Intersection Points

Solving the quadratic equation yields \(x = 3.94\) and \(x = -4.37\). Substitute these back into the line equation \(y = \frac{2-x}{3}\) to find the corresponding \(y\)-values. This gives us intersection points approximately at \((3.94, -0.65)\) and \((-4.37, 2.12)\).

Step 6: Check Solutions Graphically

Inspect the graphical plot ensuring that these calculated intersection points align with the points where the line crosses the circle.

Key Concepts

Understanding a System of Equations

A system of equations consists of two or more equations that are solved together. When dealing with a system like the one presented, we seek to find values of the variables that satisfy all the equations simultaneously.
This type of problem often involves a graphical or analytical solution, where we find where the curves or lines intersect on a graph.

• The solutions to the system are the points of intersection.
• For this specific system, the equations are a circle and a line.

To solve a system graphically, we plot each equation on the same set of axes and observe where they meet. These meeting points give us the solutions for the variables involved.

Circle Equation Explained

The circle equation, in standard form, is given as \(x^2 + y^2 = r^2\), where \(r\) is the radius. Here, the equation \(x^2 + y^2 = 25\) represents a circle centered at the origin with a radius of 5.
The points on this circle satisfy the equation because the sum of their squared distances from the origin equals the square of the radius, 25.

• Key points include where the circle intersects the x-axis and y-axis, which occurs at (5, 0), (0, 5), (-5, 0), and (0, -5).
• These key points help in sketching the complete circle.

Understanding the circle equation is essential for identifying how the circle behaves on a coordinate plane.

Linear Equation Insights

A linear equation represents a line in its simplest form. The given linear equation\(x + 3y = 2\) can be rewritten in slope-intercept form \(y = mx + b\), making it easier to graph. In this setup:

• Find intercepts which can be done by setting \(x = 0\) to find the \(y ext{-intercept}\). This gives \(y = \frac{2}{3}\).
• Set \(y = 0\) to find the \(x ext{-intercept}\), giving \(x = 2\).

Once these points are marked on the graph, you can draw the line passing through them. The slope \(-\frac{1}{3}\) reveals how steep the line is, and the direction it moves across the plane.

The Concept of Intersection Points

Intersection points are where two graphs meet or cross each other. In the context of solving a system of equations graphically, these points are the solutions that satisfy both equations simultaneously.
To find the intersection points:

• Graphically, these can be observed visually where the circle and line intersect on a graph.
• Analytically, you can substitute one equation into another and solve for the variables.

For example, you can substitute \(y = \frac{2-x}{3}\) into the circle equation \(x^2 + y^2 = 25\) to find the x-values of intersection points. Then, substitute back to find corresponding y-values.
The results from calculations should be verified with a graph to ensure accuracy of the plotted curves, confirming solutions like the points (3.94, -0.65) and (-4.37, 2.12). These solutions indicate where both the line and circle share points on the plane.