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Chapter 13: Problem 2

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

Short Answer

Expert verified
The graph of \(2x + y = 10\) is a line. The graph of \(x^2 + y^2 = 25\) is a circle.

Step by step solution

01

Identify the First Equation Type

The first equation given is \( 2x + y = 10 \). This equation is in the standard form \( Ax + By = C \), which represents a linear equation because both \( x \) and \( y \) are to the power of 1. Therefore, the graph of this equation is a line.
02

Identify the Second Equation Type

The second equation given is \( x^2 + y^2 = 25 \). This equation is in the form of \( x^2 + y^2 = r^2 \), where \( r \) is the radius. This form represents a circle where the center is at the origin \((0, 0)\) and the radius \( r \) is 5. Thus, the graph is a circle.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Equations
Linear equations are a fundamental part of algebra, representing straight lines when graphed on a coordinate plane. The general form of a linear equation is given by \( Ax + By = C \),where \( A \), \( B \), and \( C \) are constants, and \( x \) and \( y \) are variables raised to the power of 1. This form is commonly referred to as the **standard form** of a linear equation.
A few key characteristics of linear equations include:
  • The highest power of the variables is 1, which makes them linear.
  • The graph produces a straight line.
  • The solutions to these equations can visually be seen as points on this line.
  • You can also express these equations in **slope-intercept form**, \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
Linear equations are crucial for modeling real-world situations where the relationship between two variables is constant. Understanding them opens doors to more complex algebraic functions.
Standard Form
The **standard form** of equations provides a conventional way to express linear equations, usually written as \( Ax + By = C \).This form can be especially helpful because it's easy to rearrange into different forms, such as slope-intercept form.
Here's how standard form is useful:
  • It clearly displays the constant ratio, which helps to identify parallel and perpendicular lines.
  • This form makes it easy to solve systems of equations using methods like substitution or elimination.
  • Understanding standard form makes converting equations into linear functions more intuitive.
  • Despite its simplicity, standard form can represent both horizontal and vertical lines when either \( A \) or \( B \) is zero.
Mastery of the standard form is crucial in algebra as it serves as a stepping stone to exploring more advanced concepts like systems of equations and inequalities.
Equation of a Circle
The equation of a circle can appear in several forms, but the most common is the **standard form** given by \( x^2 + y^2 = r^2 \),where \( r \) is the radius and the center is assumed to be at the origin, \( (0, 0) \). This formula is derived from the Pythagorean Theorem.
A few important points about the equation of a circle include:
  • It represents a set of points that are all the same distance \( r \) from a specific center point.
  • If the center is not at the origin, the equation becomes \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center.
  • The value of \( r \) is always positive, as it's a measure of distance.
  • Circles graph as rounded shapes, different from lines or curves with varying slopes.
Understanding the equation of a circle is vital in geometry and helps in exploring conic sections and more intricate algebraic curves.

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Most popular questions from this chapter

Write the equation of a circle with a diameter whose endpoints are at \((-5,4)\) and \((7,-3)\)

Write each equation in standard form, if it is not alreacty so, and graph it. If the graph is a circle, give the coordinates of its center and its radius. If the graph is a parabola, give the coordinates of its vertex. $$ y=-4(x-4)^{2}-4 $$

Solve each system of equations for real values of x and y. $$ \left\\{\begin{array}{l} 3 x-y=-3 \\ 25 y^{2}-9 x^{2}=225 \end{array}\right. $$

Solve each system of equations by elimination for real values of x and y. $$ \left\\{\begin{array}{l} 9 x^{2}-7 y^{2}=81 \\ x^{2}+y^{2}=9 \end{array}\right. $$

Find the coordinates of the vertex and the direction in which each parabola opens. A. \(y=8(x-3)^{2}+6\) B. \(x=8(y-3)^{2}+6\)
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