Rectangular Coordinate System X-axis Y-axis (0,0) Origin Quadrant II Quadrant I Quadrant III Quadrant IV

Rectangular Coordinate System X-axis Y-axis (0,0) Rectangular Coordinate System (0,5) (-5,4) (1,3) (6, -3) (-2,-4)

Linear Equations in Two Variable A linear equation in two variables is an equation of the form Ax + By = C where A, B, C are real numbers and A & B are both not zero Examples of linear equation in two variables : 3x + 4y =23 X=6 7z + 4y =16 X + 9y = 0 The graph of any linear equation in two variables is a straight line

Solutions of Linear Equations in Two Variables Example: 2x + y = 3 Some Solutions Example: y = 3x - 1 Some Solutions x y x y 3 -1 1 1 1 2 2 -1 2 5 -1 5 -1 - 4

Graphing Linear Equations in Two Variables 5 4 3 2 1 -2 -3 -4 -5 y -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 x Graph: 2x + y = 3 (0, 3) 3 1 -1 x y 2 5 (1, 1) (2, -1)

Graphing Linear Equations in Two Variables The graph of 2x + y = 3 is a line. The solution set of a linear equation in two variables is all the points that lie on the line of its graph 5 4 3 2 1 -2 -3 -4 -5 y -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 x

X-intercepts and Y-intercepts The x-intercept is the point where the line intersects the x-axis What is the y-coordinate of the x-intercept? The y-intercept is the point where the line intersects the y-axis What is the x-coordinate of the y-intercept? 5 4 3 2 1 -2 -3 -4 -5 y -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 x 2x + y = 3

Y - Intercepts The y – intercept is the y – value when x = 0. When using a graph, the y – intercept is where the graph crosses the y – axis. The y – intercept of the line to the right is … To find the y-intercept, let x=0 in the given equation and solve for y. Then (0,y) is the y-intercept.

X - Intercepts The x – intercept is the x – value when y = 0. When using a graph, the x – intercept is where the graph crosses the x – axis. x – intercept is 2. The x – intercept of the line to the right is … To find the x-intercept, let y=0 in the given equation and solve for x. Then (x,0) is the x-intercept

y x Special Cases Graph: y = 3 This is the set of points that have a y-coordinate of 3. (x, 3) for all x. 5 4 3 2 1 -2 -3 -4 -5 y -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 x The Graph of y= 3 is a horizontal line

y x Special Cases Graph: x = 5 This is the set of points that have a x-coordinate of 5. (5, y) for all y. 5 4 3 2 1 -2 -3 -4 -5 y -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 x The graph of x= 5 is a vertical line

Graphing ax + by = c What about 3x + 4y = –12 ? Step 1: Put equation in slope – intercept form This means solve for y. Step 2: Plot the y – intercept. Step 3: Starting at the y – intercept, use the slope to find other points. A slope of means Down 3, Right 4…or…Up 3, Left 4 (the 3 or 4 can be negative, but not both).