Algebra: Slope of a Line and Graphing Lines

Slope of a Line

• The slope of a line can be calculated using the formula: (y2 - y1) / (x2 - x1)
• The slope is a measure of how steep a line is

Slope-Intercept Form

• The slope-intercept form of a linear equation is: y = mx + b
• m represents the slope and b represents the y-intercept
• Example: y = 2x - 3, where m = 2 and b = -3

Graphing Lines

• To graph a vertical line, the equation is x = a, where a is a constant
• To graph a horizontal line, the equation is y = b, where b is a constant
• Example: Graphing x = 2 and y = 3

Slope-Intercept Method

• To graph a line using the slope-intercept method, start with the y-intercept and use the slope to find another point on the line
• Example: Graphing y = 3x - 2

Standard Form

• The standard form of a linear equation is: ax + by = c
• Example: 2x - 3y = 6
• To graph a line in standard form, find the x and y-intercepts

Point-Slope Form

• The point-slope form of a linear equation is: y - y1 = m(x - x1)
• Example: Find the equation of the line passing through the point (2, 5) with a slope of 3
• Convert point-slope form to slope-intercept form by distributing the slope and adding/subtracting terms to both sides

Parallel and Perpendicular Lines

• Parallel lines have the same slope
• Perpendicular lines have slopes that are negative reciprocals of each other
• Example: Find the equation of the line passing through the point (3, -2) and parallel to the line 2x + 5y - 3
• Example: Find the equation of the line passing through the point (-4, -3) and perpendicular to the line 3x - 4y + 5

Slope of a Line

• Calculated using the formula: (y2 - y1) / (x2 - x1)
• Measured by how steep a line is

Slope-Intercept Form

• Equation: y = mx + b
• m represents the slope and b represents the y-intercept
• Example: y = 2x - 3, where m = 2 and b = -3

Graphing Lines

• Vertical line equation: x = a, where a is a constant
• Horizontal line equation: y = b, where b is a constant
• Example: x = 2 and y = 3

Slope-Intercept Method

• Start with the y-intercept and use the slope to find another point on the line
• Example: y = 3x - 2

Standard Form

• Equation: ax + by = c
• Example: 2x - 3y = 6
• Graph by finding x and y-intercepts

Point-Slope Form

• Equation: y - y1 = m(x - x1)
• Example: Find the equation of the line passing through (2, 5) with a slope of 3
• Convert to slope-intercept form by distributing the slope and adding/subtracting terms to both sides

Parallel and Perpendicular Lines

• Parallel lines have the same slope
• Perpendicular lines have slopes that are negative reciprocals of each other
• Example: Find the equation of the line passing through (3, -2) and parallel to 2x + 5y - 3
• Example: Find the equation of the line passing through (-4, -3) and perpendicular to 3x - 4y + 5