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.
![### Solving the Equation
**Example 5)**
\[ 2x - 3y = 6 \]
### Detailed Explanation of the Graph
The provided graph is a standard Cartesian coordinate plane with both the x-axis (horizontal) and y-axis (vertical) labeled from -10 to 10. The origin (0,0) is at the intersection of the axes. Both axes are marked with unit intervals.
**Key Features of the Graph:**
- The x-axis extends horizontally from -10 to 10.
- The y-axis extends vertically from -10 to 10.
- Each unit on both axes is marked by a small dot, providing a grid for plotting points accurately.
- The arrows at the end of each axis indicate that they extend infinitely in both the positive and negative directions.
### Additional Annotations
There are handwritten annotations on the graph:
- The number \(2\) is written under the graph, suggesting a calculation or specific point.
- The number \((5)\) is written next to the number \(2\), possibly indicating a point or step in solving the equation.
### Purpose of the Graph
To plot the equation \(2x - 3y = 6\) on the given coordinate plane, one would typically:
1. Find two or more points that satisfy the equation.
2. Plot these points on the coordinate plane.
3. Draw a line through the points to represent the graph of the equation.
### Steps to Plot the Equation
1. **Solve for y in terms of x:**
\[ 2x - 3y = 6 \]
\[ -3y = -2x + 6 \]
\[ y = \frac{2x - 6}{3} \]
2. **Generate Values:**
- **When \(x = 0\):**
\[ y = \frac{2(0) - 6}{3} = \frac{-6}{3} = -2 \]
(Point: \( (0, -2) \))
- **When \(x = 3\):**
\[ y = \frac{2(3) - 6}{3} = \frac{6 - 6}{3} = 0 \]
(Point: \( (3, 0) \))
3. **Plot Points:**
- \( (0, -2) \)
- \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9da2a76a-87f7-4e93-94c9-a00561324ff3%2Fa0113e66-8314-4e2d-b8a3-a98387955db4%2F6ac994j_reoriented.jpeg&w=3840&q=75)
![### Graphing Inequalities: Example Problem
#### Problem 9: Graph the Inequality
\[ -3x - 4y \leq 12 \]
#### Explanation:
To graph the inequality \(-3x - 4y \leq 12\), follow these steps:
1. **Rewrite the inequality as an equation**:
\[ -3x - 4y = 12 \]
2. **Find the intercepts**:
- **x-intercept**: Set \(y = 0\) and solve for \(x\).
\[ -3x - 4(0) = 12 \]
\[ -3x = 12 \]
\[ x = -4 \]
So, the x-intercept is \((-4, 0)\).
- **y-intercept**: Set \(x = 0\) and solve for \(y\).
\[ -3(0) - 4y = 12 \]
\[ -4y = 12 \]
\[ y = -3 \]
So, the y-intercept is \((0, -3)\).
3. **Plot the intercepts on a graph**:
- Plot the point \((-4, 0)\).
- Plot the point \((0, -3)\).
4. **Draw the boundary line**:
- Connect the points \((-4, 0)\) and \((0, -3)\) with a straight line. Since the inequality is non-strict (\(\leq\)), the line should be solid.
5. **Test a point to determine the region to shade**:
- Choose a test point not on the line, such as \((0, 0)\).
- Substitute \((0, 0)\) into the inequality:
\[ -3(0) - 4(0) \leq 12 \]
\[ 0 \leq 12 \]
This is true, so the region containing \((0, 0)\) should be shaded.
6. **Shade the appropriate region**:
- Shade the region of the graph that includes the origin \((0, 0)\).
#### Graph Explanation
The graph consists of:
- **Axes**: A set of x and y axes ranging from -10 to](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9da2a76a-87f7-4e93-94c9-a00561324ff3%2Fa0113e66-8314-4e2d-b8a3-a98387955db4%2Fte3mnyu_reoriented.jpeg&w=3840&q=75)
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