Name: The Coordinate Plane and Linear Equations Algebra Date: We use the Cartesian Coordinate plane to locate points in two-dimensional space. We can do this b measuring the directed distances the point lies awa from a horizontal and vertical ais. The first eercise and diagram will review some major features of the coordinate plane. Eercise #: (a) On the diagram, the coordinates of point A are shown State the coordinates of points B, C, and D. B A ( 3, 5 ) (b) What are the coordinates of the origin, point O? II I (c) If a point is plotted in the first quadrant, I, then what is true about the signs of both its and coordinates? III C O IV D It is etremel important for ever Algebra student to be able to quickl and accuratel plot points when given in coordinate form. Practice this skill in Eercise #. Eercise #: Given the points A(, ), B( 3, 4 ), C (, 5 ), and D( 4, 6) the grid given below and state the quadrant that each point lies in., plot and label them on QUADRANTS A: B: C: D: The Arlington Algebra Project, LaGrangeville, NY 540
Graphing a Linear Equation When we plot an equation, we are creating a picture, called a graph, of all coordinate pairs (, ) that are solutions to the equation. Graphs allow the Algebra student to quickl see man solutions to an equation. When we plot multiple points, sometimes these points will form a linear relationship; that is, when plotted, the solutions will form a line. The general form of a linear equation is shown below. GENERAL FORM OF A LINEAR EQUATION IN TWO VARIABLES A + B = C where A, B, and C are an real numbers such that A 0 and B 0. The following eercises will illustrate the important skill of plotting such linear equations. Eercise #3: Consider the linear relationship given b the equation = 4 3. (a) Create a table of values for this linear relationship. (b) Graph the linear equation on the aes below. 4 3 (, ) - 0 Keep in mind the big idea when graphing an equation, linear or otherwise: GRAPHING SOLUTIONS TO EQUATIONS If an ordered pair (, ) satisfies the equation (makes it a true statement), then it falls on the graph of that equation. Likewise, if a point (, ) falls on the graph of an equation, then it lies in the solution set of that equation. Thus a graph represents a picture of all solutions to an equation. The Arlington Algebra Project, LaGrangeville, NY, 540
Eercise #4: Consider the linear equation + 4 = 0. (a) Solve this equation for so that it ma be entered into our graphing calculator. (b) Fill in the -chart below b creating a table on our calculator. Y (, ) (c) Graph the linear equation on the aes below. The following sets of eercises will tr to reinforce the big idea of graphing b having ou work more with ordered pairs and linear equations. Eercise #5 Given the linear equation = 4 3 answer the following questions. (a) Does the point (, 5 ) satisf this equation? Justif. (b) Does the point (, 5 ) lie on the graph of this linear equation? Eplain. Eercise #6 Is the ordered pair ( 8,) a solution to the equation = 3? Justif. The Arlington Algebra Project, LaGrangeville, NY, 540
Eercise #7 Given the equation satisfies the equation. = 3, find the value of b given the fact that the point ( 0, b ) Eercise #8 Find the value of a such that the point (, 0) a lies on the graph of = + 8. Eercise #9 The graph of the equation 3 = 4 is shown at the right. Which of the following ordered pairs is not a solution to this equation? Eplain our choice. () (, ) (3) ( 3, 4) () ( 0, ) (4) (, 5 ) Eplanation: The Arlington Algebra Project, LaGrangeville, NY, 540
Name: The Coordinate Plane and Linear Equations Algebra Homework Date: Skills. On the set of aes to the right, fill the quadrant numbers in the boes.. State which quadrant each of the following points lies. ( 3, ) (, 3 ) ( 7, 5) ( 4, 6) O 3. State the coordinates of the origin, O. 4. Fill in the -chart below for the linear function = without the use of our calculator. Then, graph the equation on the aes to the right. - (, ) - 0 5. For each linear equation shown below, solve the equation for in terms of. (a) 3 + = (b) 3 = (c) 4 + = 8 The Arlington Algebra Project, LaGrangeville, NY 540
For problems 6 through 8, use our answers from problem 5 and our graphing calculator to create,-charts and plots of the linear equations. 6. 3 + = Y (, ) 7. 3 = Y (, ) 8. 4 + = 8 Y (, ) The Arlington Algebra Project, LaGrangeville, NY, 540
Reasoning 9. Two equations are shown below. One is linear and one is not. Which is the linear one and wh is it linear? = + = + 6 3 0. Consider the linear equation 5 + = 3. (a) Does the point (, 6) lie in the solution set of this equation? Justif. (b) Does the point (, 6) lie on the graph of this linear equation? Eplain. (c) For what value of a will the point (, a) fall on the graph of this linear equation?. The graph of a linear equation is shown on the aes below. Answer the following questions based on this graph. (a) For what value of b is the point ( 0, b ) a solution to this linear equation? Justif. (b) For what value of a will the point ( a, 3) be a solution to this linear equation? Justif. The Arlington Algebra Project, LaGrangeville, NY, 540