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.

This quiz covers the fundamental concepts of linear equations, including slope, intercept, and various forms of line equations. Understand how to identify parallel and perpendicular lines, and learn the techniques for graphing them accurately.