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**
**Objective:**
Write the equation of the line perpendicular to \( y - 6 = -\frac{2}{3}(x - 7) \) and passing through the point (8, -1). Write in the format \( y = mx + b \).
### Instructions
1. Identify the slope of the given line.
2. Determine the slope of the perpendicular line.
3. Use the point-slope form of the line equation with the given point (8, -1) and the slope found in step 2.
4. Convert the equation to the format \( y = mx + b \).
**Help Section**
**Understanding the Problem:**
To find the equation of the line that is perpendicular to a given line, you must first understand that the slopes of two perpendicular lines are negative reciprocals of each other.
Given line: \( y - 6 = -\frac{2}{3}(x - 7) \)
- The slope of this line is \(-\frac{2}{3}\).
Slope of the perpendicular line:
- The perpendicular slope (m) is the negative reciprocal of \(-\frac{2}{3}\), which is \(\frac{3}{2}\).
Given point:
- The point through which the perpendicular line passes is (8, -1).
**Steps:**
1. Use the slope-intercept form: \( y = mx + b \).
2. Substitute the point (8, -1) into the equation to find b.
**Calculation:**
\[ -1 = \frac{3}{2}(8) + b \]
\[ -1 = 12 + b \]
\[ b = -1 - 12 \]
\[ b = -13 \]
**Final Equation:**
\[ y = \frac{3}{2}x - 13 \]
### Answer Box
Using the calculations above, the perpendicular line's equation in the format \( y = mx + b \) is:
\[ y = \frac{3}{2}x - 13 \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F20047395-754f-4b4f-b55f-051358973abe%2F31d10915-30a8-458e-8d17-3ea5594f510c%2Fflm8zjq.jpeg&w=3840&q=75)
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