4.1 Introduction to Linear Equations in Two Variables A linear equation in two variables can be put in the form (called standard form): where A, B, and C are real numbers and A and B are not zero

4.1 Introduction to Linear Equations in Two Variables Table of values (try to pick values such that the calculation of the other variable is easy): x y 6 2 3 4

4.1 Introduction to Linear Equations in Two Variables Points: (2, 3) 2 is the x-coordinate, 3 is the y-coordinate Quadrants: II x<0 and y>0 I x>0 and y>0 III x<0 and y<0 IV x>0 and y<0

4.2 Graphing by Plotting and Finding Intercepts The graph of any linear equation in two variables is a straight line. Note: Two points determine a line. Graphing a linear equation: Plot 3 or more points (the third point is used as a check of your calculation) Connect the points with a straight line.

4.2 Graphing by Plotting and Finding Intercepts x y 6 2 3 4 Graph:

4.2 Graphing by Plotting and Finding Intercepts Finding the x-intercept (where the line crosses the x-axis): let y = 0 and solve for x Finding the y-intercept (where the line crosses the y-axis): let x = 0 and solve for y Note: the intercepts may be used to graph the line.

4.2 Graphing by Plotting and Finding Intercepts If y = k, then the graph is a horizontal line: If x = k, then the graph is a vertical line:

4.2 Graphing by Plotting and Finding Intercepts x y -3 2 4 Example: Graph the equation.

4.3 The Slope of a Line The slope of a line through points (x1,y1) and (x2,y2) is given by the formula:

4.3 The Slope of a Line A positive slope rises from left to right. A negative slope falls from left to right.

4.3 The Slope of a Line If the line is horizontal, m = 0. If the line is vertical, m = undefined.

4.3 The Slope of a Line Finding the slope of a line from its equation Solve the equation for y. The slope is given by the coefficient of x Example: Find the slope of the equation.

4.4 The Slope-Intercept Form of a Line Standard form: Slope-intercept form: (where m = slope and b = y-intercept)

4.4 The Slope-Intercept Form of a Line Example: Put the equation 2x + 3y = 6 in slope-intercept form, determine the slope and intercept, then graph. Since b = 2, (0,2) is a point on the line. Since , go down 2 and across 3 to point (3,0) a second point on the line, then connect the two points to draw the line.

4.4 The Slope-Intercept Form of a Line x y 2 3 Example: Graph the equation.

4.5 Writing an Equation of a Line Standard form: Definition is now changed as follows: A, B, and C must be integers with A > 0 Slope-intercept form: (where m = slope and b = y-intercept) Point-slope form: for a line with slope m going through point (x1, y1).

4.5 Writing an Equation of a Line Example: Find the equation of a line going through the point (2,5) with slope = 3. Express your answer in slope-intercept form. Start with the point-slope equation: Solve for y to get in slope intercept form:

4.5 Writing an Equation of a Line Example: Find the equation of a line going through the points (-3,5) and (0,3). Express your answer in standard form. Solve for the slope: Use slope intercept form & multiply by the LCD:

4.6 Parallel and Perpendicular Lines Parallel lines (lines that do not intersect) have the same slope. Perpendicular lines (lines that intersect to form a 90 angle) have slopes that are negative reciprocals of each other. Horizontal lines and vertical lines are perpendicular to each other

4.6 Parallel and Perpendicular Lines Example: Determine if the lines are parallel, perpendicular or neither: get the slope of each line the slopes are negative reciprocals of each other so the lines are perpendicular

4.6 Parallel and Perpendicular Lines Example: Find the equation in slope-intercept form of a line passing through the point (-4,5) and perpendicular to the line 2x + 3y = 6 (solve for y to get slope of line) (take the negative reciprocal to get the  slope)

4.6 Parallel and Perpendicular Lines Example (continued): Use the point-slope form with this slope and the point (-4,5) In slope intercept form: