Writing and Graphing Linear Equations

Presentation on theme: "Writing and Graphing Linear Equations"— Presentation transcript:

1 Writing and Graphing Linear Equations
Linear equations can be used to represent relationships.

2 Writing Equations and Graphing
These activities introduce rates of change and defines slope of a line as the ratio of the vertical change to the horizontal change. This leads to graphing a linear equation and writing the equation of a line in three different forms.

3 Linear equation – An equation whose solutions form a straight line on a coordinate plane. Collinear – Points that lie on the same line. Slope – A measure of the steepness of a line on a graph; rise divided by the run.

4 A linear equation is an equation whose solutions fall on a line on the coordinate plane. All solutions of a particular linear equation fall on the line, and all the points on the line are solutions of the equation. Look at the graph to the left, points (1, 3) and (-3, -5) are found on the line and are solutions to the equation.

5 If an equation is linear, a constant change in the x-value produces a constant change in the y-value. The graph to the right shows an example where each time the x-value increases by 2, the y-value increases by 3.

6 The equation y = 2x + 6 is a linear equation because it is the graph of a straight line and each time x increases by 1 unit, y increases by 2

7 Real world example The graph (c = 5x + 10) at the left shows the cost for Company A cell phone charges. What does Company A charge for 20 minutes of service?

8 Graphing equations can be down several different ways
Graphing equations can be down several different ways. Tables can be used to graph linear equations by simply graphing the points from the table.

9 Complete the table below, then graph and tell whether it is linear.

10 Can you determine if the equation is linear
Can you determine if the equation is linear? The equation y = 2x + 3 is a linear equation because it is the graph of a straight line. Each time x increases by 1 unit, y increases by 2.

11 Slope Rate of change

12 Slope of a line is its rate of change
Slope of a line is its rate of change. The following example describes how slope (rate of change) is applied. Rate of change is also know as grade or pitch, or rise over run. Change is often symbolized in mathematics by a delta for which the symbol is the Greek letter: Δ

13 Finding slope (rate of change) using a graph and two points.

14 If an equation is linear, a constant change in the x-value corresponds to a constant change in the y-value. The graph shows an example where each time the x-value increases by 3, the y-value increases by 2.

15 Slopes: positive, negative, no slope (zero), undefined.

16 Remember, linear equations have constant slope
Remember, linear equations have constant slope. For a line on the coordinate plane, slope is the following ratio. This ratio is often referred to as “rise over run”.

17 Find the slope of the line that passes through each pair of points.
(1, 3) and (2, 4) (0, 0) and (6, -3) (2, -5) and (1, -2) (3, 1) and (0, 3) (-2, -8) and (1, 4)

18 Graphing a Line Using a Point and the Slope Graph the line passing through (1, 3) with slope 2.

19 Given the point (4, 2), find the slope of this line?
To make finding slope easier, find where the line crosses at an x and y junction.

20 Finding Slope from a Graph Use the graph of the line to determine its slope. Choose two points on the line (-4, 4) and (8, -2). Count the rise over run or you can use the slope formula. Notice if you switch (x1, y1) and (x2, y2), you get the same slope:

21 Use the graph to find the slope of the line.

22 Slope-intercept Form y = mx + b

23 Y = mx + b Slope-intercept Form
An equation whose graph is a straight line is a linear equation. Since a function rule is an equation, a function can also be linear. m = slope b = y-intercept Slope-intercept Form Y = mx + b (if you know the slope and where the line crosses the y-axis, use this form)

24 b = -7, so the y-intercept is -7
For example in the equation; y = 3x + 6 m = 3, so the slope is 3 b = +6, so the y-intercept is +6 Let’s look at another: y = 4/5x -7 m = 4/5, so the slope is 4/5 b = -7, so the y-intercept is -7 Please note that in the slope-intercept formula; y = mx + b the “y” term is all by itself on the left side of the equation. That is very important!

25 WHY? If the “y” is not all by itself, then we must first use the rules of algebra to isolate the “y” term. For example in the equation: 2y = 8x + 10 You will notice that in order to get “y” all by itself we have to divide both sides by 2. After you have done that, the equation becomes: Y = 4x + 5 Only then can we determine the slope (4), and the y-intercept (+5)

26 And “b” (the y-intercept) = +9
OK…getting back to the lesson… Your job is to write the equation of a line after you are given the slope and y-intercept… Let’s try one… Given “m” (the slope) = 2 And “b” (the y-intercept) = +9 All you have to do is plug those values into y = mx + b The equation becomes… y = 2x + 9

27 Let’s do a couple more to make sure you are expert at this.
Given m = 2/3, b = -12, Write the equation of a line in slope-intercept form. Y = mx + b Y = 2/3x – 12 ************************* One last example… Given m = -5, b = -1 Y = -5x - 1

28 Slope-intercept form of an equation
Given the slope and y-intercept, write the equation of a line in slope-intercept form. 1) m = 3, b = -14 2) m = -½, b = 4 3) m = -3, b = -7 4) m = 1/2 , b = 0 5) m = 2, b = 4 Slope-intercept form of an equation Y = mx + b

29 Using slope-intercept form to find slopes and y-intercepts The graph at the right shows the equation of a line both in standard form and slope-intercept form. You must rewrite the equation 6x – 3y = 12 in slope-intercept to be able to identify the slope and y-intercept.

30 Using slope-intercept form to write equations, Rewrite the equation solving for y = to determine the slope and y-intercept. 3x – y = 14 -y = -3x + 14 y = 3x – 14 or 3x = y + 14 3x – 14 = y x + 2y = 8 2y = -x + 8 y = -1x + 4 2