Linear Equations - Algebra

Definition: The slope is calculated as the rise over the run.
- Positive Slope: Indicates a line rising from left to right.
- Examples of Slope Calculation:
  - Going up 4 units (rise), right 3 units (run): ( m = \frac{4}{3} )
  - Going down 3 units (rise), right 5 units (run): ( m = -\frac{3}{5} )
Slope Values:
- A line at 45 degrees has a slope of 1.
- Less steep lines have smaller slopes (e.g., ( \frac{1}{2} )), while steeper lines have greater slopes (e.g., 2).
- Horizontal Lines: Slope is 0.
- Vertical Lines: Slope is undefined.
Slope Calculation Using Two Points:
- Formula: ( m = \frac{y_2 - y_1}{x_2 - x_1} )
- Example: From points ( (2, 5) ) and ( (5, 14) ):
  - Calculation: ( m = \frac{14 - 5}{5 - 2} = \frac{9}{3} = 3 )

X-Intercept: The point on the x-axis where ( y = 0 ).
- Example: ((3, 0)) and ((-5, 0))
Y-Intercept: The point on the y-axis where ( x = 0 ).
- Examples: ((0, 4)) leads to ( b = 4 ) and ((0, -3)) leads to ( b = -3 ).
Summary:
- The x-intercept is where ( y = 0 ); the y-intercept is where ( x = 0 ).

Parallel Lines:
- Have the same slope.
- Formula: If line 1 has slope ( m_1 ), line 2 also has slope ( m_2 ) such that ( m_1 = m_2 ).
Perpendicular Lines:
- Intersect at 90 degrees.
- The slope of a perpendicular line is the negative reciprocal of the other:
  - If ( m_1 ) = ( \frac{3}{4} ), then ( m_2 ) = ( -\frac{4}{3} ).

Graphing in Slope-Intercept Form (Example: ( y = 2x - 4 )):
- Identify y-intercept ( b = -4 ); plot this point.
- Using the slope ( 2 ) (rise of 2, run of 1), plot subsequent points.
Graphing in Standard Form (Example: ( 3x - 2y = 6 )):
- Find x-intercept (set ( y = 0 )) and y-intercept (set ( x = 0 )).
Graphing in Point-Slope Form (Example: ( y - 3 = 2(x - 2) )):
- Identify the slope (2) and the point (2,3) to plot.
- Use slope to identify additional points.

Horizontal Line: Formed when ( y = k ); slope is 0.
- Example: ( y = 3 ) gives a horizontal line at y = 3.
Vertical Line: Formed when ( x = h ); slope is undefined.
- Example: ( x = 4 ) gives a vertical line at x = 4.

Multiple-choice questions focusing on identifying slopes and intercepts.
Free response problems involving calculations and graphing, enhancing understanding of linear equations.

This document provides an in-depth exploration of linear equations, essential for understanding algebra and preparing for exams.

There are three primary forms of writing linear equations, each serving specific purposes:

Slope-Intercept Form:
- Written as: ( y = mx + b )
- ( m ) represents the slope of the line, determining its steepness and direction. A positive slope indicates the line rises from left to right, while a negative slope indicates it falls.
- ( b ) represents the y-intercept, the point where the line crosses the y-axis. It is essential for graphing the equation accurately.
Standard Form:
- Written as: ( Ax + By = C )
- In this format, ( A ), ( B ), and ( C ) are coefficients which can be integers and must be such that ( A ) is non-negative. This form is beneficial for quickly identifying intercepts and can be rearranged into other forms easily.
Point-Slope Form:
- Written as: ( y - y_1 = m(x - x_1) )
- This form is particularly useful when you know the slope and a specific point on the line. It allows for easy calculation of any additional points along the line by manipulating the equation.

Definition: The slope is calculated as the rise over the run, reflecting the rate of change between the variables.
Positive Slope: A line with a positive slope indicates an increase; as the x-values increase, the corresponding y-values also increase.
Negative Slope: A line with a negative slope indicates a decrease; as the x-values increase, the corresponding y-values decrease.

Example 1: Going up 4 units (rise), right 3 units (run): ( m = \frac{4}{3} )
Example 2: Going down 3 units (rise), right 5 units (run): ( m = -\frac{3}{5} )

A line at 45 degrees has a slope of 1, indicating equal rise and run.
Less steep lines have smaller slopes (e.g., ( \frac{1}{2} )), while steeper lines have greater slopes (e.g., 2).
Horizontal Lines: Have a slope of 0, indicating no change in y-values as x-values change.
Vertical Lines: Have an undefined slope because they do not exhibit a change in x-values.

Formula: ( m = \frac{y_2 - y_1}{x_2 - x_1} )
Example: From points ( (2, 5) ) and ( (5, 14) ), the slope calculation is ( m = \frac{14 - 5}{5 - 2} = \frac{9}{3} = 3 ).

X-Intercept: The point on the x-axis where ( y = 0 ), represented as ( (x, 0) ).
- Examples: ((3, 0)), ((-5, 0)).
Y-Intercept: The point on the y-axis where ( x = 0 ), represented as ( (0, y) ).
- Examples: ((0, 4)) which leads to ( b = 4 ), and ((0, -3)) leading to ( b = -3 ).

The x-intercept is where ( y = 0 ); the y-intercept is where ( x = 0 ), crucial for graphing purposes and solving equations.

Definition: Parallel lines have the same slope but different y-intercepts.
- Formula: If line 1 has slope ( m_1 ), line 2 also has slope ( m_2 ) such that ( m_1 = m_2 ).

Definition: Perpendicular lines intersect at a 90-degree angle.
- Slope Relationship: The slope of a perpendicular line is the negative reciprocal of the other.
- Example: If ( m_1 ) = ( \frac{3}{4} ), then ( m_2 ) = ( -\frac{4}{3} ).

Graphing in Slope-Intercept Form (Example: ( y = 2x - 4 )):
- Identify y-intercept ( b = -4 ); plot the point (0, -4).
- Using the slope (2) (rise of 2, run of 1), plot subsequent points from the y-intercept.
Graphing in Standard Form (Example: ( 3x - 2y = 6 )):
- Find the x-intercept (set ( y = 0 )) to find the point where the line crosses the x-axis.
- Find the y-intercept (set ( x = 0 )) to find the crossing point on the y-axis.
Graphing in Point-Slope Form (Example: ( y - 3 = 2(x - 2) )):
- Identify the slope (2) and the specific point (2,3) to plot.
- Use the slope to identify additional points by moving in the rise/run direction from the starting point.

Horizontal Line: Formed when ( y = k ); it will have a slope of 0.
- Example: ( y = 3 ) gives a horizontal line at y = 3.
Vertical Line: Formed when ( x = h ); it does not have a defined slope, as it cannot be represented numerically.
- Example: ( x = 4 ) gives a vertical line at x = 4.

Multiple-choice questions focusing on identifying slopes and intercepts based on given equations and graphs.
Free response problems involving calculations of slopes, intercepts, and graphing lines, enhancing overall understanding of linear equations and their properties.