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.
![**Problem Statement:**
*Write the equation of the line \( y - \frac{3}{4}x + 2 \) in Standard Form.*
**Explanation:**
In algebra, the Standard Form of a linear equation is written as:
\[ Ax + By = C \]
Where \( A \), \( B \), and \( C \) are integers, and \( A \) should be non-negative.
**Solution:**
Given the equation:
\[ y - \frac{3}{4}x + 2 \]
First, we need to rearrange it into the Standard Form. We start by getting all terms involving \(x\) and \(y\) on one side of the equation, and the constant term on the other side.
Initial equation:
\[ y - \frac{3}{4}x + 2 = 0 \]
Move \(-\frac{3}{4}x\) to the other side:
\[ y + 2 = \frac{3}{4}x \]
Isolate the \(y\)-term by subtracting 2 from both sides:
\[ y = \frac{3}{4}x - 2 \]
Next, we eliminate the fraction by multiplying every term by 4 (the denominator of the fraction):
\[ 4y = 3x - 8 \]
Now, rearrange the equation to the Standard Form \(Ax + By = C\):
\[ 3x - 4y = 8 \]
**Final Answer:**
\[ 3x - 4y = 8 \]
This is the equation of the line in Standard Form.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F010da4e9-01a7-4c8c-bd06-26050acf6ced%2F6157f73f-cdec-4de7-a6de-4f1f7d522aea%2F3yvij3_processed.jpeg&w=3840&q=75)
Step by step
Solved in 2 steps with 1 images
