The equation of the line often will be written initially in this form The equation of the line often will be written initially in this form. You may be asked to write the answer in this form as well. The standard form of a line is Ax +By=C, where B and C are integers, and A is a nonnegative integer. (A and B are not both zero) Standard Form of a Line The slope-intercept form of a line is y=mx+b, where m is a real number representing the slope and b is a real number representing the y intercept. Slope-Intercept Form This form is used to graph the equation. This form may also be used to write the equation given the slope and the y-intercept. You may be asked to write the answer in this form. Next Slide This form is used to write the equation of a line given a point on the line and it’s slope. It could also be used if given two points. With two points, the slope can be found using the slope formula from the previous section. Point-Slope Form The point-slope form is y – y1=m(x – x1) where (x1, y1) are the coordinates of a point on a line and m is a real number representing the slope of the line.

Example 1. Write the equation of the line that has the indicated slope and contains the indicated point. Express final equation in standard form. Point Slope Determine which form to start with. Since we have the slope and a point, use the point-slope form. Substitute point and slope into the formula and simplify. Remember to clear fractions by multiplying both sides by the LCD. Remember to leave in standard form as requested in the directions. Your Turn Problem #1 Write the equation of the line that has the indicated slope and contains the indicated point. Express final equation in standard form. , Answer:

Example 2. Write the equation of the line that contains the indicated pair of points. Express final equation in standard form. , To get an equation of a line, we need one point and the slope. Since we have two points, use the slope formula to find the slope. Now we have a point and the slope. Use the point-slope formula and simplify. Your Turn Problem #2 Write the equation of the line that contains the indicated pair of points. Express final equation in standard form. , Answer:

Example 3. Write the equation of the line that has the indicated slope (m) and y intercept (b). Express final equations in slope-intercept form. , Since we have the slope and the y intercept, use the slope intercept formula (y = mx +b). Substitute the given slope and intercept into the equation. Your Turn Problem #3 Write the equation of the line that has the indicated slope (m) and y intercept (b). Express final equations in slope-intercept form. , Answer:

Express final equations in standard form. Example 4. Write the equation of the line that satisfies the given conditions. Express final equations in standard form. x intercept of 2 and y intercept of –3 The intercepts represent two points. Now that we have the slope and a point, use the point-slope formula. First use the slope formula to find the slope. Then use the point-slope formula to find the equation of the line. A must be positive Your Turn Problem #4 Write the equation of the line that satisfies the given conditions. Express final equations in standard form. x intercept of –1 and y intercept of –3

Notes: 1. Example of a graph with a positive slope. 2. Example of a graph with a negative slope. 3. The graph of a horizontal line has a slope of 0. The equation of any horizontal line is y=k where k is the y intercept. 4. The slope of a vertical line has a slope which is undefined. The equation of any vertical line is x=c where c is the x intercept.

Parallel Lines Two lines are parallel if there slopes are equal and they have different y-intercepts. Example of two parallel lines: (The lines never intersect.) More Notes Next Slide Perpendicular Lines Two lines are perpendicular if their slopes are negative reciprocals of each other. This means that the slopes have opposite signs and are reciprocals of each other. Example of two perpendicular lines: The lines intersect at a 90° angle.

Contains the point (3,–2) and is parallel to the y axis Example 5. Write the equation of the line that satisfies the given conditions. Express final equations in standard form. Contains the point (3,–2) and is parallel to the y axis Solution: y axis x axis A line that is parallel to the y axis is a vertical line. The equation of a vertical line is x=c where c is the x intercept. Your Turn Problem #5 Write the equation of the line that satisfies the given conditions. Express final equations in standard form. Contains the point (2,–1) and is parallel to the x axis Answer:

Contains the point (–3, –1) and is perpendicular to the y axis Example 6. Write the equation of the line that satisfies the given conditions. Contains the point (–3, –1) and is perpendicular to the y axis Solution: y axis x axis A line that is perpendicular to the y axis is a horizontal line. The equation of a horizontal line is y=k where k is the y intercept. Your Turn Problem #6 Write the equation of the line that satisfies the given conditions. Express final equations in standard form. Contains the point (2,5) and is perpendicular to the x axis.

Writing an equation in slope-intercept form. Once an equation is written in slope-intercept form, the points can be obtained by either choosing values for x or using the slope and the y-intercept to find points. Procedure: 1. Multiply both sides by the LCD (if there are denominators other than 1). 2. Isolate the y-term on the LHS. Write the RHS in in descending order. 3. Divide all terms by the coefficient of the y-term. Move the x-term to the RHS. Divide all terms by 2. 2 Answer: Your Turn Problem #7 Write 3x – 5y =15 in slope-intercept form. Answer:

Solution: Example 8. Write the equation of the line that satisfies the given conditions. Express final equations in slope-intercept form. Since the lines are parallel, the slopes will be the same. Rewrite the line given in slope-intercept form to determine the slope. Now we have the slope and a point. We can use the point-slope formula. Simplify and write answer in slope-intercept form. Your Turn Problem #8 Write the equation of the line that satisfies the given conditions. Express final equations in slope-intercept form.

Solution: Example 9. Write the equation of the line that satisfies the given conditions. Express final equations in standard form. Now use the point-slope formula, simplify and write in standard form. Find slope of line given. Since the desired line is perpendicular to the given line, use m=4. Your Turn Problem #9 Write the equation of the line that satisfies the given conditions. Express final equations in standard form.

Using the Slope and y-Intercept to Graph a Line Many students prefer to graph using the slope and y-intercept because it requires very little calculations. The slope of a line is defined to be the change in y divided by the change in x. The saying “rise over run” may be useful here. Procedure: Graphing lines using the y-intercept and slope. Step 1. Write the equation in the form y = mx + b. This is called slope-intercept form of a line. Step 2. Use b to plot the y-intercept. This is the point where the line crosses the y-axis. It can be written as (0,b). Step 3. Use the slope to find two other points from the y-intercept. The numerator of the slope gives the number of units to go up or down. The denominator of the slope gives the number of units to go left or right. , from the y-intercept (or any point), up 2 units, right 5 units. from the y-intercept, down 2 units, left 5 units. , from the y-intercept (or any point), down 3 units, right 7 units. from the y-intercept, up 3 units, left 7 units. Step 4. Draw the line through the points.

• • • Step 1. Already in slope-intercept form. Step 2. Plot the y-intercept. Since b=-1, the y-intercept is (0,-1). Step 3. Use the slope to find two other points. So from the y-intercept (or any point), go up 2 units then right 5 units. • go down 2 units then left 5 units. • • Step 4. Draw the line through the points. Next Slide

• Your Turn Problem # 10 y-intercept: (0,1) (down 2, right 3) or (up 2, left 3) or

• • Step 1. Write in slope-intercept form. Step 2. Plot the y-intercept. Since b=2, the y-intercept is (0,2). Step 3. Use the slope to find two other points. 5 Next Slide • • So from the y-intercept (or any point), go down 2 units then right 5 units or up 2 units then left 5 units. Step 4. Draw the line through the points.

• Your Turn Problem #11 y-intercept: (0,8) (down 4, right 3 or up 4, left 3) Written by Bob Ramirez 11-25-06