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.
![### Graph the Line 3y - x = -6
Use the provided graph to plot the line described by the equation 3y - x = -6. Below is an explanation on how we can approach graphing this equation:
1. **Transform the Equation**:
Rearrange the equation into slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
\[
3y - x = -6 \\
\Rightarrow 3y = x - 6 \\
\Rightarrow y = \frac{1}{3}x - 2
\]
Here, the slope (m) is \( \frac{1}{3} \) and the y-intercept (b) is -2.
2. **Plot the Y-Intercept**:
Start by plotting the y-intercept on the graph. Here, the y-intercept is at (0, -2). Mark this point on the vertical axis.
3. **Use the Slope to Find Another Point**:
The slope of \( \frac{1}{3} \) means that for every 1 unit we move horizontally to the right, we move up by \( \frac{1}{3} \) units.
From the point (0, -2), move 3 units to the right and 1 unit up to find another point on the line, which is (3, -1).
4. **Draw the Line**:
Using a ruler or a straight-edge tool, draw a line through the points (0, -2) and (3, -1) extending it in both directions.
### Graph Description:
The graph displays a standard coordinate plane with x and y axes ranging from -5 to 5.
- The horizontal axis (x-axis) is labeled from -5 to 5.
- The vertical axis (y-axis) is labeled from -5 to 5.
### Tool Instructions:
At the bottom of the graph interface, there are multiple tool options:
- Line tool: Select to draw straight lines.
- Parabola tool: Select to draw parabola shapes.
- Angle tool: Select to draw angles.
- Circle tool: Select to draw circles.
- Clear All button: Use to clear the graph and start over.
Ensure you use the straight line tool to plot the line \( y = \frac{1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcebbde0b-56df-456f-b45f-057baf1d3a0f%2Fb0b743c9-5893-45cd-986c-1329bc13b252%2Fm9drlc_processed.png&w=3840&q=75)
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