Linear Equations Basics

Slope and Intercept

• Slope (m): Measures the steepness of a line; calculated as the change in y over the change in x (rise/run).
• Y-Intercept (b): The point where the line crosses the y-axis (where x=0).
• Slope-Intercept Form: The equation of a line can be represented as ( y = mx + b ).

Equation of a Line

• Standard Form: ( Ax + By = C ), where A, B, and C are constants, and A ≥ 0.
• Point-Slope Form: ( y - y_1 = m(x - x_1) ), where ( (x_1, y_1) ) is a point on the line and m is the slope.
• Intercept Form: ( \frac{x}{a} + \frac{y}{b} = 1 ), where a is the x-intercept and b is the y-intercept.

Parallel and Perpendicular Lines

• Parallel Lines: Have the same slope (m1 = m2) but different y-intercepts; never intersect.
• Perpendicular Lines: Slopes are negative reciprocals (m1 * m2 = -1); if one slope is m1, the other is -1/m1.

Graphing Linear Equations

• Plotting Points: Choose values for x, calculate corresponding y values using the line equation.
• Intercepts: Identify x-intercept (set y=0) and y-intercept (set x=0) to plot two points.
• Drawing the Line: Connect the plotted points with a straight line extending in both directions.

Angle Between Two Straight Lines

• Formula: The angle ( \theta ) between two lines with slopes m1 and m2 is given by: [ \tan(\theta) = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| ]
• Interpretation: If (\theta = 0), the lines are parallel; if (\theta = 90^\circ), the lines are perpendicular.

Slope and Intercept

• Slope (m): Represents how steep a line is, calculated as the ratio of vertical change to horizontal change (rise/run).
• Y-Intercept (b): The specific point on the y-axis where the line intersects, occurring at x = 0.
• Slope-Intercept Form: A linear equation can be expressed as ( y = mx + b ), succinctly showing the slope and y-intercept.

Equation of a Line

• Standard Form: Write a linear equation in the format ( Ax + By = C ), where A, B, and C are constants and A must be non-negative.
• Point-Slope Form: An equation can be represented as ( y - y_1 = m(x - x_1) ), utilizing a known point ( (x_1, y_1) ) on the line and its slope.
• Intercept Form: The equation ( \frac{x}{a} + \frac{y}{b} = 1 ) highlights the x-intercept (a) and y-intercept (b).

Parallel and Perpendicular Lines

• Parallel Lines: Share identical slopes (m1 = m2) ensuring they do not cross at any point.
• Perpendicular Lines: Their slopes are related as negative reciprocals (m1 * m2 = -1), meaning if one slope is m1, the other becomes -1/m1.

Graphing Linear Equations

• Plotting Points: Generate a table of values for x and derive corresponding y values based on the line's equation for accurate plotting.
• Intercepts: Locate x-intercept by setting y to zero and y-intercept by setting x to zero; these provide critical points for graphing.
• Drawing the Line: After plotting points from both intercepts, extend a straight line through these points across the graph.

Angle Between Two Straight Lines

• Angle Formula: The angle ( \theta ) between two lines, with slopes m1 and m2, is determined by: [ \tan(\theta) = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| ]
• Interpretation: An angle of ( \theta = 0 ) indicates the lines are parallel, while ( \theta = 90^\circ ) confirms they are perpendicular.