Rectangular Coordinate System In algebra, we used number lines to plot numbers and equations and inequalities of 1 variable (x = -3, x one-dimensional) x - axis y - axis origin (0,0) To examine equations involving 2 variables, we graph on a rectangular (Cartesian) coordinate system (y = x, y = x - 1 => two-dimensional) On a plane, each point is a pair of numbers => (1,2); (-2,-3); (3,-1) (plot on board or have students plot) (1,2) (-2,-3) (3,-1)

Quadrants and Finding Coordinates Coordinates like (2,3) are called ordered pairs and are of the form (x,y), where x is the x-coordinate, and y is the y-coordinate Graphs can be divided into 4 quadrants –Quadrant I => both coordinates are positive –Quadrant II => 1st-coordinate negative / 2nd-coordinate positive –Quadrant III => both coordinates are negative –Quadrant II => 1st-coordinate positive / 2nd-coordinate negative I II III IV R: (, ) B: (, ) G: (, ) P: (, )

Solutions of Equations To determine if an ordered pair is a solution of an equation, we use the 1st number in the pair to replace the variable that occurs 1st alphabetically The solution of an equation in 2 variables (typically x and y) is an ordered pair which when substituted into the equation give a true statement Because of this, we can generate ordered-pair solutions to equations Are (-1,2) and (-5,4) solutions of 6q + 3p = 9? xy Are other points on the line between (-1,-3) and (2,3) solutions to y = 2x -1? Show that (-1,-3) and (2,3) are solutions to y = 2x – 1 and graph

Graphs of Linear Equations A linear equation in 2 variables is an equation that can be written in the form Ax + By = C where A and B are not both 0 (standard form) The graphs of linear equations are straight lines To graph, find at least 2 ordered pairs, plot those points, and draw a line connecting and through the points Graph -x + 2y = 0 and 4x + 2y = -6 below xy = ½ xy The point where x = 0 is the y-intercept (and vice versa) In a linear equation solved for y (y = mx + b), the graph passes through the y-intercept at (0,b)

Linear Equations Graph the following and identify the y-intercepts y + 5x = 1 x + 2y = 6 x x Which of the following are linear equations: 3x 2 + 4y = 1 7x + 3 = 2y 7y = 14 3x = -27

Identifying Intercepts A y-intercept of a graph is a point where the graph intersects the y-axis (this is also the point where x = 0) An x-intercept of a graph is a point where the graph intersects the x-axis (this is also the point where y = 0) Find the x and y intercepts for the following… Red: Blue: Green: Purple: 2x + 5y = 10: y = 4 – x 2 :

More complex graphs Graph the following –y = 3 – x 2 using numbers from -2 to 2 for x –y = |x + 1| xy = 3 – x 2 y = |x + 1| -2y = 3 – 4 = -1y = |-2 + 1| =

Linear equations are often used to model real-life applications EXAMPLE –Suppose Bert started selling his paperclips online through SesamE- Bay. If he incurs an up-front setup fee of $10 and makes a profit of a half-dollar for every paperclip sold, what linear equation could be used to model his earnings? E = (if E represents profit and x represents items sold) –Determine Bert’s earnings if selling 0,5,10,20,40,100 paperclips (and graph the model) Linear Equations Applied

Graphs are often used to show trends over time or a range By identifying points on line graphs and their coordinates, you can interpret specific information given in the graph – How many rushing yards did the Hokies have in 2005? In 2006? – How many passing yards did the Hokies have in 2003? – For the period given, what is the biggest total of rushing yards in a season for the Hokies and in what season? – Around how many more passing yards did VT have than rushing yards in 2007? Interpreting Graphs