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Remember to read the textbook before attempting to do your homework. Section 3.1: Introduction to Linear Equations in 2 Variables Section 3.2: Graphing by Plotting Points and Finding Intercepts y Rectangular Coordinate System aka Cartesian coordinate system, aka xy-plane The x-axis (horizontal axis) and y-axis (vertical axis) divide the plane into four quadrants (regions): The graph of an ordered pair, (x, y), is a point where the 1st number is called the x-coordinate, the 2nd number is called the y-coordinate. Quadrant 2 (–,+) Origin Things to note: Quadrant 3 The origin is the point where the x-and y-axis meet. (–,–) The coordinates of the origin are given by (0, 0). The points on the x-axis or y-axis do not belong to any quadrant. The order of the numbers is very important, e.g. (2, 5) is not the same as (5, 2) Example 1: a. Is (3, 5) a solution of 2x – 3y = –9 ? Quadrant 1 (+,+) (0, 0) x Quadrant 4 (+,–) b. Is (5, 3) a solution of 2x – 3y = –9 ? Example 2: Graph the following set of ordered pairs and label the coordinates of the points: y {(0, 0), (–1, 3), (0, –5), (5, 0), (3.5, 4), (–3, –4)} Friendly Reminder: Ordered pairs are written in the form (x, y). To graph an ordered pair, first start from the origin, and then . . . x 1. See how many units you should move to the left or right (this is determined by the x-coordinate) 2. See how many units you should move up or down (this is determined by the y-coordinate) Math 60 || Beginning Algebra || Cerritos College – Pg 1 – Chapter 3 Lecture Notes by Maria Torres Standard Form A linear equation in 2 variables written in the form Ax + By = C, is said to be in standard form, where A, B, and C are real numbers (A and B not both 0). In English: Fix the equation so you have the x-term + y-term = constant term. Also, please make sure that you have integer coefficients. Question: I know that the graph of an ordered pair is a point, but what does the graph of a linear equation look like? Answer: A straight line. Question: How do you graph of a line? Answer: Oh that’s easy! Just find 2 ordered pair solutions of the equation, graph them, and then connect the points. Question: Is there an easy way to find ordered pair solutions? Answer: Of course there is! Just ... 1. Solve the equation for y, then 2. Create an xy-table, and then 3. Pick a value for x and find the corresponding value for y. ☺ Ex 3: Graph by plotting at least 3 points for each graph. You may not use a calculator to answer. a. –6x + 2y = 4 y x b. 4x = –3y – 9 y x Math 60 || Beginning Algebra || Cerritos College – Pg 2 – Chapter 3 Lecture Notes by Maria Torres Question: Is there a different way to obtain the graph of a line? Answer: Yes, there is! We can also graph a line by using the x-intercept and the y-intercept. y Intercepts of a Line Since the graph x-intercept: The point where the line intercepts the x-axis. To find the x-intercept: Let y = 0, and then Solve for x. intercepts the x-axis at 1, then Note: The x-intercept is of the form (x1, 0) x-int = (1, 0) y-intercept: The point where the line intercepts the y-axis. To find the y-intercept: Let x = 0, and then Solve for y. Since the graph intercepts the Note: The y-intercept is of the form (0, y1) y-axis at –2, then y-int = (0, –2) x Example 4: What is the x-intercept? What is the y-intercept? Use the x-intercept and y-intercept to graph the line. Label the coordinates of the intercepts. y a. –4x + 5y = 20 x y b. 14 – 4y = 7x x Bored or just looking for more fun? Then work on the attached Xtra Practice Sheet (intercepts) Math 60 || Beginning Algebra || Cerritos College – Pg 3 – Chapter 3 Lecture Notes by Maria Torres Xtra Practice: Using intercepts to graph linear equations Sample: Use the intercepts to graph –2x – 3y = 6. On your graph, please label the coordinates of the intercepts. Solution: Goal #1: Goal #2: Goal #3: Find the x-intercept Find the y-intercept Use the intercepts to graph the line. To find the x-intercept: Let y = 0, then solve for x. To find the y-intercept: Let x = 0, then solve for y. –2x – 3y = 6 –2x – 3y = 6 –2x – 3(0) = 6 –2(0) – 3y = 6 –2x = 6 –3y = 6 x = –3 (–3, 0) (0, –2) y = –2 P.S. x-intercept: (–3, 0) y-intercept: (0, –2) Don’t forget to label the intercepts Directions: What is the x-intercept? What is the y-intercept? Use the x-intercept and y-intercept to graph the line. Label the coordinates of the intercepts y y x a. 5x – 6y = –30 Math 60 || Beginning Algebra || Cerritos College x b. 8y – 4x = –32 – Pg 4 – Chapter 3 Lecture Notes by Maria Torres y y x c. –4 = – 4y + x x d. 5x = 35 – 7y y y x e. 18 = 2y x f. –y = 4x + 6 (Solutions will be posted on the Math 60 page) Math 60 || Beginning Algebra || Cerritos College – Pg 5 – Chapter 3 Lecture Notes by Maria Torres Vertical and Horizontal Lines The graph of any equation of the form The graph of any equation of the form x = x1 y = y1 is a vertical line that intercepts the x-axis at x1. is a horizontal line that intercepts the y-axis at y1. Moral of the story: If an equation does not contain BOTH x and y variables, then the graph of the equation will be either a vertical line or a horizontal line. Example 5: Graph x = 2 Example 6: Graph y = –3 Psst psst . . . The graph of x = 2 is a vertical line that intercepts the x-axis at positive 2. Psst psst . . . the graph of y = –3 is a horizontal line that intercepts the y-axis at negative 3. Example 7: Graph the lines. a. x + 3 = 0 b. y – 8 = 0 c. 3y + 10 = –2 d. 2x – 14 = –2 (end of Section 3.1/3.2 combo) Math 60 || Beginning Algebra || Cerritos College – Pg 6 – Chapter 3 Lecture Notes by Maria Torres