GRAPHING LINEAR EQUATIONS

GRAPHING LINEAR EQUATIONS MSJC ~ San Jacinto Campus Math Center Workshop Series Janice Levasseur

Basic Definitions • Axes – perpendicular number lines • x-axis – horizontal number line • y-axis – vertical number line • Origin – the point of intersection of the axes • Ordered pair – a number pair (x, y), coordinate, point • Abscissa – first coordinate, x • Ordinate – second coordinate, y

Quadrant II Quadrant I Quadrant III Quadrant IV

Graphs On a number line, each point is the graph of a number 0 2 On a plane, each point is the graph of a number pair ordered pair: (x, y) (1,2)

Ex: Plot (3, 5) Ex: Plot (-3, 4) Ex: Plot (-4, -2)

An ordered pair is a solution to a 2-variable equation if a true statement results when the equation is evaluated at the ordered pair. Ex: Show (4, 5) and (-2, 2) are solutions to y = ½ x + 3 y = ½ x + 3 y = ½ x + 3 5 ½ (4) + 3 2 ½ (-2) + 3 2 -1 + 3 5 2 + 3 2 2 5 5 True  (4, 5) is a solution! True  (-2, 2) is a solution!

Ex: Plot (4, 5) and (-2, 2) on the same set of axes. Notice a straight line connect the points (solutions)

Connecting all the points that are solutions to an equation will result in a straight line. • In other words, the graph of the line connecting the points (solutions) represents the solution set of the equation. • Since the graph of the solution set is represented by a straight line, the equation is called a linear equation. • More formally, an equation in 2 variables, where the exponents of the variables are 1, is called a linear equation. Ax + By = C or y = mx + b, where A, B, C, m, and b are constants and A and B are not both 0

Graphing a Linear Equation(Plotting Points) • Solve the equation for one of the variables, usually y • Pick a value for x, plug it in, & solve for y • Repeat at least two more times • Plot the points on the same set of axes • Connect the dots with a straight line

Ex: Graph 6x – 3y = 3 Solve for a variable: y x y = 2x - 1 y 6x – 3y = 3 0 y = 2(0) - 1 -1 -6x -6x 1 y = 2(1) - 1 1 - 3y = 3 – 6x -3 -3 ½ y = 2(½) - 1 0 y = - 1 + 2x y = 2x - 1 We have identified 3 solutions to the equation: (0, -1) (1, 1) ( ½ , 0)

Plotting the three solutions/points we get: (0, -1) (1, 1) ( ½ , 0) The solution points lie on a straight line. Every point on this line is a solution to the equation 6x – 3y = 3! 1 1

Your turn to try a few

We can always plot points to graph linear equations • However, plotting points could be tedious, (especially for “messy” equations) • There must be other ways to plot linear equations . . .

Graphing using Intercepts • Consider a linear equation of the form Ax + By = C • The y-intercept is the point in which the graph of the line crosses the y-axis, (0, b) • To find the y-intercept, let x = 0 and solve for y • The x-intercept is the point in which the graph of the line crosses the x-axis, (a, 0) • To find the x-intercept, let y = 0 and solve for x

Ex: Graph 2x – 3y = 6 using intercepts y-int: let x = 0 x-int: let y = 0 2(0) – 3y = 6 2x – 3(0) = 6 – 3y = 6 2x = 6 y = - 2 x = 3 (3, 0) (0, - 2) 1 1

Slope-Intercept Form Consider the linear equation Ax + By = C Solving for y, we get an equation of the form y = mx + b, where m and b are constants y = mx + b is called the slope-intercept form because b is the “intercept” (y-intercept (0, b)) and m is the “slope”

When the slope is negative (m < 0), the line slants down from left to right When the slope is positive (m > 0), the line slants up from left to right

Graphing using the slope and y-intercept Given the slope-intercept form, we can identify the slope, m, and the y-intercept, (0, b) To graph an equation of a line, given the slope-intercept form, start by plotting the y-intercept Then use the slope to identify at least 2 more solutions of the equation (i.e. solution points)

Recall, y = mx + b, where m and b are numbers, is the slope-intercept form of a linear equation. Ex: Find the slope-intercept form of x + 5y = 10 To find the slope-intercept form, we need to solve for y y = m x + b x + 5y = 10 m = - 1/5 -x -x b = 2 5y = - x + 10 1 5 5 (0, 2) is the y-int y = (-1/5) x + 2

Graph using Slope-Intercept form: y = (-1/5) x + 2 -1 Slope m = and y-int = (0, 2) 5 Plot (0, 2) Next, use the slope rise = - 1 down 1 run = 5 right 5 2 2 Note: -a/b = a/(-b)  m = 1/(-5)  up 1, left 5

Ex: Graph 2x + 3y = -9 solve for y to use the slope-int form 2x + 3y = - 9 y = m x + b -2x -2x -2 m = 3y = - 2x - 9 3 Negative slope  line slants 3 3 down from left to right y = (-2/3)x – 3 rise = -2  down 2 run = 3  right 3 b = - 3  (0, -3) is the y-int  START HERE

Graph: y = (-2/3)x - 3 -2 Slope m = and y-int = (0, -3) Plot (0, -3) 3 Next, slope rise = -2 down 2 run = 3 right 3 or rise = 2 up 2 run = -3 left 3 2 2

Special Lines The graph of y = b is a horizontal line with y-intercept (0, b) y is always b no matter what x is The graph of x = a is a vertical line with x-intercept (a, 0) x is always a no matter what y is

Ex: Graph 7x + 63 = 0 7x + 63 = 0 7x = - 63 x = - 9 x is always – 9 no matter what y is  vertical 3 3

Ex: Graph 12y = 48 12y = 48 y = 4 y is always 4 no matter what x is  horizontal 3 3

Horizontal lines have slope m = 0 No vertical change  rise = 0  m = 0 Vertical lines have undefined slope, m = undefined No horizontal change  run = 0  m undefined rise m = run

To Graph a Linear Equation: • Plot points: pick nice x and solve for y  (x, y), find at least 3 solutions • Intercepts: • y-int: set x = 0, solve for y  (0, y) • x-int: set y = 0, solve for x  (x, 0) • Slope-Intercept: y = mx + b • Start with y-int (0, b) • Use slope m to plot at least 2 more points • Only one variable: • y = b  horizontal line with y-int (0, b) • x = a  vertical line with x-int (a, 0)

GRAPHING LINEAR EQUATIONS