1. Chapter 7 Section 1 The Cartesian Coordinate System and Linear Equations in Two Variables

2. Learning Objective Plot points in the Cartesian (Rectangular) Coordinate System. Determine whether an ordered pair is a solution to a linear equation. Key Vocabulary: graph, Cartesian coordinate system, rectangular coordinate system, quadrants, x-axis, y- axis, origin, coordinates, ordered pairs, linear equation in two variables, collinear

3. Plot Points in the Cartesian (Rectangular) Coordinate System A graph shows the relationship between two variables in an equation. Ordered pairs are the x- and y-coordinates of a point are placed in parentheses, with the x- coordinate listed first.

4. Plot Points in the Cartesian (Rectangular) Coordinate System IVIII III Origin (0,0) y-axis - vertical x-axis horizontal The two intersecting axes form four quadrants that are numbered I, II, III and IV. Origin is where the x and y axes intersect Negative Positive

5. Plot Points in the Cartesian (Rectangular) Coordinate System B C A (0,0) Example: Plot each point on the same axes. A(4,4) B(-2,3) C(-4,-2) D(0,2) E(2,-2) F(-3,0) D F E

6. Plot Points in the Cartesian (Rectangular) Coordinate System B C A (0,0) Example: List the ordered pairs for each point. A = (3,2) B = (0,-3) C = (-2,-2) D = (-1,0) E = (2,-4) F = (-4,2) D F E

7. Determine whether an Ordered Pair is a Solution to a Linear Equation A linear equation in two variables is an equation that can be put in the form, ax + by = c, where a, b, and c are real numbers. The graphs of equations of the form ax + by = c are straight lines. This is why they are called linear. The adjective straight is not needed to describe lines. All lines are straight. It is used to emphasize the shape of the graph. A set of point that are in a straight line are said to be collinear.

8. Determine whether an Ordered Pair is a Solution to a Linear Equation (-1.-5) (2,4) (0,0) Example: Determine whether the three points appear to be collinear. (-1,-5) (0,-2) (2,4) Yes the line is collinear. (0,-2)

9. Determine whether an Ordered Pair is a Solution to a Linear Equation (-2.1) (2,4) (0,0) Example: Determine whether the three points appear to be collinear. (-3,-2) (-2,1) (2,4) No, the line is not collinear. (-3,-2)

10. Determine whether an Ordered Pair is a Solution to a Linear Equation Example: Determine which of the following ordered pairs satisfy the equation. x + 2y = 7 (3,2), (4,-1), (-1,4), (-3,5) x + 2y = 7 3 + 2(2) = 7 3 + 4 = 7 7 = 7 TRUE x + 2y = 7 4 + 2(-1) = 7 4 + -2 = 7 2 ≠ 7 FALSE x + 2y = 7 -1 + 2(4) = 7 -1 + 8 = 7 7 = 7 TRUE x + 2y = 7 -3 + 2(5) = 7 -3 + 10 = 7 7 = 7 TRUE

11. Determine whether an Ordered Pair is a Solution to a Linear Equation (4.-1) (3,2) (0,0) (-1,4) Example:. x + 2y = 7 (3,2) True (4,-1) False (-1,4) True (-3,5) True (-3,5)

12. Remember With ordered pairs the order does matter. x is always first. (x, y) We graph solutions to linear equations in one variable on a number line to give a visual representation of the solution set. We graph solutions to linear equations in two variables using ordered pairs in the Cartesian (rectangular) plane to give a visual representation of the solution set. Two distinct points completely determine a line.

13. HOMEWORK 7.1 Page 429-430: #27, 29, 37, 39