Find an equation of the line parallel to the line y- 4x 8 and contains the poir || Write your answer in point-slope form. O y = 3 – 4(x + 2) O y = 3 - (x – 2) O y = 3+(x - 2) O y = 3 + 4 (x + 2)
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.
![### Parallel Line Equation Problem
**Problem Statement:**
Find an equation of the line parallel to the line \( y - 4x = 8 \) and contains the point (-2, 3). Write your answer in point-slope form.
**Options:**
1. \( y = 3 - 4(x + 2) \)
2. \( y = 3 - \frac{1}{4}(x - 2) \)
3. \( y = 3 + \frac{1}{4}(x - 2) \)
4. \( y = 3 + 4(x + 2) \)
**Explanation:**
To solve this problem, follow these steps:
1. **Identify the Slope of the Given Line:**
The given line is \( y - 4x = 8 \), which can be rewritten in slope-intercept form as \( y = 4x + 8 \). Thus, the slope (\( m \)) of the given line is 4.
2. **Parallel Lines Have Equal Slopes:**
Any line parallel to this must also have a slope of 4.
3. **Point-Slope Form:**
Use the point-slope form of the equation of a line: \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is the given point and \( m \) is the slope.
4. **Substitute the Point and Slope:**
For the given point (-2, 3) and slope 4:
\[
y - 3 = 4(x - (-2))
\]
Simplifying, you get:
\[
y - 3 = 4(x + 2)
\]
Thus, the correct option is:
- \( y = 3 + 4(x + 2) \)
**Graph/Diagram Explanation:**
In this problem, there are no specific graphs or diagrams provided. The text involves a problem-solving approach related to linear equations. The focus is on recognizing the slope of parallel lines and correctly applying the point-slope form using provided coordinates.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F882c71c8-a3eb-4cc5-9d5e-6f9c46c87e68%2Fcb270e73-5d50-4819-ba86-57d9dbdc16de%2Fzf250j_processed.jpeg&w=3840&q=75)
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