Opener #6 (Ch. 1.1-1.2) Introduction to Lines Graph using a table 4x + 2y = 10. 2. Determine if the following are linear… a) y = xy b) 4y = 6x – 1 c) x = 23 3. Find the intercepts and graph… 2x = ¼ y – 2 4. Determine the slope between (4,3) and (4-8)

More Linear Equations Graphing linear equations from slope-intercept form and writing equations of lines.

How are lines graphed? How do you determine the equation of a line?

Graph the following equations… Try This… Graph the following equations… The line through (4, -2) with a slope of 4/3. The line through (-3, 2) with a slope of -4. The line through (5, 0) with an undefined slope.

Slope-Intercept Form Slope-Intercept Form looks like this… y = mx + b where m = slope and b is the y-intercept From this equation you can determine a point on the line and the slope…enough to graph the line!

How do I graph a linear equation? (Continued) Method 3: Slope-Intercept Form Steps: Solve the equation for y. (Isolate y) Identify the slope (m) and the y-intercept (b) Plot the y-intercept (Better be on y-axis) Use the slope to determine at least one point above and below the intercept Draw a line through the points

Example: 2x + 3y = 9 3y = -2x + 9 y = -2/3 x + 3 m = -2/3 (downhill slope) b = 3 (y-intercept) Will graph on the board now…

Try These Graph from slope-intercept form… 1) 4x + y = - 4

Horizontal and Vertical Lines Horizontal and vertical lines are easy to identify because they only have ONE variable in the equation… like x = 3 or y = -4. What causes the issue is determining which it is…horizontal or vertical… It’s the opposite of the axis represented by the variable in the equation… For example x = 3, x is the horizontal axis…then this line is the opposite - vertical

Parallel Lines To determine if lines are parallel, put them in slope-intercept form and compare their slopes and why intercepts. THE SLOPES WILL BE IDENTICAL AND INTERCEPTS DIFFERENT! All horizontal lines are parallel and all vertical lines are parallel

Perpendicular Lines To determine if lines are perpendicular, put them in slope-intercept form and compare their slopes. If the lines are perpendicular, then the product of the two slopes will be -1. So multiply (slope 1)(slope 2), if it equals -1 they are perpendicular. Horizontal lines and vertical lines are always perpendicular.

Determine if the following pairs of lines are parallel, perpendicular, or neither. 2y + 3x = 4 2. y = ¼ x + 6 4y = -6x + 2 8x + 2y = 10 3. y = 20 x = 2

What is the minimum I need to write a linear equation? So….what do you think you need to know about a line to write an equation? (Think about slope-intercept form) What two values change from line to line in that form?

Writing Equations of Lines So we must have…SLOPE AND A POINT! (May not be given exactly in this way…you may be given two points, etc.)

How do I use Point-Slope Form? Steps: Identify/calculate the slope of the line. Determine a point to use on the line Plug the values in for x1, y1, and m in to the formula: y - y1 = m(x - x1) Solve the equation for y.

Examples of Point-Slope Form Write the equation of a line with m = ½ and through the point (5, -3) Write the point-slope form y - y1 = m(x - x1) m = ½ x1 = 5 y1 = -3 (Sub in) y – (-3) = ½ (x – 5) (simplify) y + 3 = ½ x – 5/2 (solve for y) y = ½ x – 11/2

More… 2. Write the equation of a line parallel to the line y = 3x – 4 through (2, 2) y - y1 = m(x - x1) m = 3 (Same as given line because para) x1 = 2 y1 = 2 y – 2 = 3(x – 2) y – 2 = 3x – 6 y = 3x - 4

More… 3. Write the equation of a line through the points (4, 2) and (-2, 5). 4. Write the equation of a line perpendicular to y = 2/3 x + 4.

Try These! Write the equation of a line with slope of -3 through the point (0, 5). Write the equation of a line perpendicular to y = 3x – 2 through (-1, 3) Write the equation of the line through the points (4, -8) and (-6, 3).