Review of Linear Equations
Review of Linear Equations
Overview
This guide focuses on linear equations in preparation for tests.
Key forms of linear equations are discussed, along with concepts such as slope, intercepts, parallel and perpendicular lines, and graphing techniques.
Forms of Linear Equations
Slope-Intercept Form
Definition: A linear equation in slope-intercept form is expressed as:
y = mx + b
Where:
m = Slope
b = y-intercept
Standard Form
Definition: A linear equation in standard form is expressed as:
Ax + By = C
Where:
A, B, and C are coefficients
x and y are variables
Point-Slope Form
Definition: A linear equation in point-slope form is written as:
y - y1 = m(x - x1)
Where:
m = slope
(x1, y1) = A specific point on the line
Understanding Slope
Definition of Slope
Slope is defined as the ratio of the change in the vertical coordinate (rise) to the change in the horizontal coordinate (run):
ext{slope} = rac{ ext{rise}}{ ext{run}}
Examples of Slope Calculation
Example 1:
If from point 1 (P1) to point 2 (P2) you move up 4 and right 3:
ext{slope} = rac{4}{3} (positive slope)
Example 2:
If you move down 3 (negative rise) and right 5:
ext{slope} = rac{-3}{5} (negative slope)
Characteristics of Slope
Positive slope: Line rises from left to right.
Negative slope: Line falls from left to right.
Slope of 0: Horizontal line.
Undefined slope: Vertical line.
Special Angle Slope Values
45-degree upward: Slope = 1
Steeper upward lines: Slope increases (e.g., slope = 2)
Downward 45-degree angle: Slope = -1
General observations:
Horizontal line = slope of 0
Vertical line = slope is undefined.
Calculating Slope Between Two Points
Formula:
If points are (x1, y1) and (x2, y2), then:
ext{slope} = rac{y2 - y1}{x2 - x1}
Example calculation:
Point 1: (2, 5)
Point 2: (5, 14)
ext{slope} = rac{14 - 5}{5 - 2} = rac{9}{3} = 3
X-Intercepts and Y-Intercepts
X-Intercept
Definition: The x-intercept is the point at which y = 0.
Example: For the point (3, 0), the x-intercept is 3.
Y-Intercept
Definition: The y-intercept is the point at which x = 0.
Example: For the point (0, 4), the y-intercept is 4.
Summary of Intercepts
X-intercept: Value of x when y = 0.
Y-intercept: Value of y when x = 0.
Identifying Intercepts from Given Points
Given points: (2, 5), (-3, 0), (1, 2), (0, 6)
X-intercept: Point (-3, 0) means x-intercept is -3.
Y-intercept: Point (0, 6) means y-intercept is 6.
Parallel and Perpendicular Lines
Parallel Lines
Definition: Parallel lines have the same slope.
Notation: m1 = m2
Example: If line 1 has a slope of 2, then line 2 also has a slope of 2.
Perpendicular Lines
Definition: Perpendicular lines intersect at right angles (90 degrees).
Relationship: The slope of one line is the negative reciprocal of the other.
If line 1 has slope m1, then line 2 has slope m2:
m1 imes m2 = -1
Example: If m1 = rac{3}{4}, then m2 = - rac{4}{3}.
Graphing Techniques
Graphing from Slope-Intercept Form
Given: y = 2x - 4
Identify slope and y-intercept:
Slope: 2, y-intercept: -4
Steps:
Plot y-intercept (0, -4).
Use slope (rise/run = 2/1) to find another point.
Graphing from Standard Form
Given: 3x - 2y = 6
Find x-intercept: Set y = 0 -> x = 2 (point: (2, 0))
Find y-intercept: Set x = 0 -> y = -3 (point: (0, -3))
Steps:
Plot intercepts (2, 0) and (0, -3).
Connect points to form line.
Graphing Other Forms
Point-Slope Form
Given: y - 3 = 2(x - 2)
Identify point (2, 3) and slope = 2.
Steps:
Plot point (2, 3).
Use slope of 2 to find additional points.
Graphing Constant Lines
Horizontal line for y = 3 at y = 3 (slope = 0).
Vertical line for x = 4 at x = 4 (slope is undefined).
Practice Problems
Use multiple choice and free response problems to reinforce understanding of concepts discussed.
Example Problem: Identify the graph for y = 2x - 3
Slope: 2 (line rises)
Y-Intercept: -3 (touches y-axis at -3).
Find correct graph based on slope direction and intercept location.
Follow similar steps in practice problems to confirm understanding of linear equations, their graphs, and properties.
Review of Linear Equations
Overview
This guide focuses on linear equations in preparation for tests.
Key forms of linear equations are discussed, along with concepts such as slope, intercepts, parallel and perpendicular lines, and graphing techniques.
Forms of Linear Equations
Slope-Intercept Form
Definition: A linear equation in slope-intercept form is expressed as:
y = mx + b
Where:
m = Slope
b = y-intercept
Standard Form
Definition: A linear equation in standard form is expressed as:
Ax + By = C
Where:
A, B, and C are coefficients
x and y are variables
Point-Slope Form
Definition: A linear equation in point-slope form is written as:
y - y1 = m(x - x1)
Where:
m = slope
(x1, y1) = A specific point on the line
Understanding Slope
Definition of Slope
Slope is defined as the ratio of the change in the vertical coordinate (rise) to the change in the horizontal coordinate (run):
ext{slope} = frac{ ext{rise}}{ ext{run}}
Examples of Slope Calculation
Example 1:
If from point 1 (P1) to point 2 (P2) you move up 4 and right 3:
ext{slope} = frac{4}{3} (positive slope)
Example 2:
If you move down 3 (negative rise) and right 5:
ext{slope} = frac{-3}{5} (negative slope)
Characteristics of Slope
Positive slope: Line rises from left to right.
Negative slope: Line falls from left to right.
Slope of 0: Horizontal line.
Undefined slope: Vertical line.
Special Angle Slope Values
45-degree upward: Slope = 1
Steeper upward lines: Slope increases (e.g., slope = 2)
Downward 45-degree angle: Slope = -1
General observations:
Horizontal line = slope of 0
Vertical line = slope is undefined.
Calculating Slope Between Two Points
Formula:
If points are (x1, y1) and (x2, y2), then:
ext{slope} = frac{y2 - y1}{x2 - x1}
Example calculation:
Point 1: (2, 5)
Point 2: (5, 14)
ext{slope} = frac{14 - 5}{5 - 2} = frac{9}{3} = 3
X-Intercepts and Y-Intercepts
X-Intercept
Definition: The x-intercept is the point at which y = 0.
Example: For the point (3, 0), the x-intercept is 3.
Y-Intercept
Definition: The y-intercept is the point at which x = 0.
Example: For the point (0, 4), the y-intercept is 4.
Summary of Intercepts
X-intercept: Value of x when y = 0.
Y-intercept: Value of y when x = 0.
Identifying Intercepts from Given Points
Given points: (2, 5), (-3, 0), (1, 2), (0, 6)
X-intercept: Point (-3, 0) means x-intercept is -3.
Y-intercept: Point (0, 6) means y-intercept is 6.
Parallel and Perpendicular Lines
Parallel Lines
Definition: Parallel lines have the same slope.
Notation: m1 = m2
Example: If line 1 has a slope of 2, then line 2 also has a slope of 2.
Perpendicular Lines
Definition: Perpendicular lines intersect at right angles (90 degrees).
Relationship: The slope of one line is the negative reciprocal of the other.
If line 1 has slope m1, then line 2 has slope m2:
m1 \times m2 = -1
Example: If m1 = frac{3}{4}, then m2 = - frac{4}{3}.
Graphing Techniques
Graphing from Slope-Intercept Form
Given: y = 2x - 4
Identify slope and y-intercept:
Slope: 2, y-intercept: -4
Steps:
Plot y-intercept (0, -4).
Use slope (rise/run = 2/1) to find another point.
Graphing from Standard Form
Given: 3x - 2y = 6
Find x-intercept: Set y = 0 -> x = 2 (point: (2, 0))
Find y-intercept: Set x = 0 -> y = -3 (point: (0, -3))
Steps:
Plot intercepts (2, 0) and (0, -3).
Connect points to form line.
Graphing Other Forms
Point-Slope Form
Given: y - 3 = 2(x - 2)
Identify point (2, 3) and slope = 2.
Steps:
Plot point (2, 3).
Use slope of 2 to find additional points.
Graphing Constant Lines
Horizontal line for y = 3 at y = 3 (slope = 0).
Vertical line for x = 4 at x = 4 (slope is undefined).
Practice Problems
Use multiple choice and free response problems to reinforce understanding of concepts discussed.
Example Problem: Identify the graph for y = 2x - 3
Slope: 2 (line rises)
Y-Intercept: -3 (touches y-axis at -3).
Find correct graph based on slope direction and intercept location.
Follow similar steps in practice problems to confirm understanding of linear equations, their graphs, and properties.