Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
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### Algebraic Equation
The equation given in point-slope form is:
\[ y - 7 = \frac{1}{3}(x + 3) \]
### Explanation
This equation represents a line in a two-dimensional Cartesian coordinate system. To understand this equation better, let's break it down:
- **Point-Slope Form of a Line:**
The standard form for the point-slope equation of a line is given by:
\[ y - y_1 = m(x - x_1) \]
Where:
- \( (x_1, y_1) \) is a point on the line.
- \( m \) is the slope of the line.
- **Components of Our Equation:**
In the given equation \( y - 7 = \frac{1}{3}(x + 3) \):
- The point \((x_1, y_1)\) on the line is \((-3, 7)\).
- Here, the equation is in the form where we subtract these coordinates from \( x \) and \( y \) respectively.
- The slope \( m \) of the line is \( \frac{1}{3} \).
This indicates that for every unit increase in \( x \), \( y \) increases by \( \frac{1}{3} \) units.
**Graphical Representation:**
- Plot the point \((-3, 7)\) on the graph.
- From this point, use the slope \(\frac{1}{3}\) to determine the next points. For example, moving 3 units to the right (forward in \( x \)-direction) and 1 unit up (forward in \( y \)-direction) from \((-3, 7)\) will help plot the next point.
- Draw a line through these points to extend in both directions, and you have the graph of the equation \( y - 7 = \frac{1}{3}(x + 3) \).
\[
\begin{array}{c|c}
x & y \\
\hline
-3 & 7 \\
0 & 8 \\
3 & 9 \\
\ldots & \ldots \\
\end{array}
\]
This table can help visualize some specific points the line](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F34bff1e1-af68-4229-b0ef-aa0aaa134e46%2Fa3fe2a69-1054-4de7-809b-33aa664c96b4%2Fyqf677_processed.jpeg&w=3840&q=75)
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