Problem 2 - Intermediate Algebra

Question

Fill in the blanks. The graph of \(2 x+y=10\) is a _____ and the graph of \(x^{2}+y^{2}=25\) is a _____.

Step-by-Step Solution

Answer

The graph of \(2x + y = 10\) is a line, and the graph of \(x^2 + y^2 = 25\) is a circle.

Step 1: Identify the Type of Equation for 2x + y = 10

The equation is in the form \(ax + by = c\), which is the standard form of a linear equation. Thus, the graph of this equation is a line.

Step 2: Identify the Type of Equation for x^2 + y^2 = 25

This equation is in the form \(x^2 + y^2 = r^2\), which represents a circle with center at the origin \((0, 0)\) and radius \(r\). Here, \(r^2 = 25\), so \(r = 5\). Hence, the graph is a circle.

Step 3: Fill in the Blanks

From the above steps, the graph of \(2x + y = 10\) is a **line**, and the graph of \(x^2 + y^2 = 25\) is a **circle**.

Key Concepts

Linear Equations

Linear equations are fundamental in mathematics and one of the simplest types of algebraic expressions. A linear equation is an equation between two variables that results in a straight line when graphed on a coordinate plane. For example, in the equation \(2x + y = 10\), the terms are combined in a way that every term is either a constant or a constant multiplied by a single variable. This is the typical format of a linear equation, which can be rearranged to the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
This form helps us understand how steep the line is and where it crosses the y-axis. A linear equation can also be presented in the standard form \(ax + by = c\), which is easy to use for finding intercepts and plotting the graph. The key characteristics of linear equations involve:

• Constant slope \(m\), which describes the steepness and direction of the line.
• A straight line graph that can extend infinitely in both directions.
• Simplicity in solving for unknown values, usually involving addition, subtraction, multiplication, or division.

Understanding linear equations is essential as it serves as the building block for more complex topics in algebra and calculus.

Circles

A circle is a perfectly round shape that is defined in a coordinate plane by its center and its radius. The equation of a circle is presented in the form \(x^2 + y^2 = r^2\), where \((x, y)\) represents any point on the circle, and \(r\) is the radius. For instance, the equation \(x^2 + y^2 = 25\) represents a circle centered at the origin, \((0, 0)\), with a radius of 5.
This specific form helps visualize where the circle sits in the coordinate plane and its size. It's crucial to recognize and understand these points about circles:

• A circle's radius is the distance from its center to any point along the edge of the circle.
• The center of a circle in its standard position is at the origin \((0, 0)\), unless specified otherwise.
• All points that lie on the circle are equidistant from the center, reflecting the geometric property of uniform distance.

Circles are widely used in geometry, and knowing how to work with their equations opens the door to understanding many natural and man-made objects and phenomena.

Graphing Equations

Graphing equations is a vital skill for visualizing and understanding mathematical relationships and data. It involves plotting points, lines, or curves on a coordinate grid to represent algebraic expressions or functions. Each point on the graph offers a visual representation of solutions to the equation. For linear equations, the process is straightforward. Starting with finding the x and y intercepts by setting each variable to zero in turn-simple calculations help identify these points.
In the case of quadratic or circular equations, like \(x^2 + y^2 = 25\), plotting can involve recognizing the shape of the graph, such as a parabola or circle, by understanding the form of the equation. To successfully graph an equation, one should keep in mind:

• The type of equation you're working with—linear, quadratic, or otherwise—this dictates the shape represented on the graph.
• Accurately identifying key points, such as intercepts, centers, or vertices, to assist in drawing the plot accurately.
• Using appropriate scales and plotting methods to ensure accuracy and clarity of the graph.

Mastering graphing techniques allows you to better explore mathematical concepts and see how equations correspond to visual representations in the real world.