Write an equation of the line and interpret the slope.
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.
![**Title: Understanding Linear Equations and Slope**
**Objective:** Write an equation of the line and interpret the slope using the provided graphs.
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**1. Interest-Free Loan**
**Graph Description:**
- The graph is titled "Interest-Free Loan."
- The x-axis represents "Payments (months)" ranging from 0 to 30.
- The y-axis represents "Balance (dollars)" ranging from 0 to 2000.
- Two key data points are indicated: (10, 1300) and (20, 800).
**Interpretation:**
- The line on the graph shows a downward trend, indicating that as payments increase, the loan balance decreases.
- To find the equation of the line, identify the slope using the formula:
\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{800 - 1300}{20 - 10} = \frac{-500}{10} = -50
\]
- Equation of the line: \( y = mx + b \)
- \( m = -50 \) (slope)
- Use one point, e.g., (10, 1300), to find \( b \):
\[
1300 = -50(10) + b \implies b = 1800
\]
- Final equation: \( y = -50x + 1800 \)
- **Slope Interpretation:** The slope of -50 indicates that for each month of payment, the balance decreases by $50.
---
**2. Canoe Rental**
**Graph Description:**
- The graph is titled "Canoe Rental."
- The x-axis represents "Time (hours)" ranging from 0 to 6.
- The y-axis represents "Total Cost (dollars)" ranging from 0 to 50.
- Two key data points are indicated: (5, 45) and (2, 30).
**Interpretation:**
- The line shows an upward trend, indicating that as time increases, the total cost increases.
- Calculate the slope:
\[
\text{slope} = \frac{45 - 30}{5 - 2} = \frac{15}{3} = 5
\]
- Equation of the line: \( y = mx + b \)
- \( m =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F394982b0-b3d4-4ad9-89c3-1de870dc122d%2Fb3b05c46-33f8-4191-909a-f3b437abcf44%2Fss9f0qj_processed.jpeg&w=3840&q=75)
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