57 graphing lines from linear equations

Linear Equations and Lines

In the rectangular coordinate system, ordered pairs (x, y)’s correspond to locations of points.


In the rectangular coordinate system, ordered pairs (x, y)’s correspond to locations of points. Collections of points may be specified by the mathematics relations between the x-coordinate and the y coordinate.


In the rectangular coordinate system, ordered pairs (x, y)’s correspond to locations of points. Collections of points may be specified by the mathematics relations between the x-coordinate and the y coordinate. The plot of points that fit a given relation is called the graph of that relation.


In the rectangular coordinate system, ordered pairs (x, y)’s correspond to locations of points. Collections of points may be specified by the mathematics relations between the x-coordinate and the y coordinate. The plot of points that fit a given relation is called the graph of that relation. To make a graph of a given mathematics relation, make a table of points that fit the description and plot them.


Example A. Graph the points (x, y) where x = –4



Example A. Graph the points (x, y) where x = –4 (y can be anything).



Example A. Graph the points (x, y) where x = –4 (y can be anything).Make a table of ordered pairs of points that fit the description x = –4.





x y –4 –4 –4 –4

Make a table of ordered pairs of points that fit the description x = –4.




x y –4 0 –4 –4 –4





x y –4 0 –4 2 –4 –4





x y –4 0 –4 2 –4 4 –4 6





x y –4 0 –4 2 –4 4 –4 6

Graph of x = –4




First degree equation in the variables x and y are equations that may be put into the form Ax + By = C where A, B, C are numbers.


First degree equation in the variables x and y are equations that may be put into the form Ax + By = C where A, B, C are numbers. First degree equations are the same as linear equations.


First degree equation in the variables x and y are equations that may be put into the form Ax + By = C where A, B, C are numbers. First degree equations are the same as linear equations. They are called linear because their graphs are straight lines.


First degree equation in the variables x and y are equations that may be put into the form Ax + By = C where A, B, C are numbers. First degree equations are the same as linear equations. They are called linear because their graphs are straight lines. To graph a linear equation, find a few ordered pairs that fit the equation.


First degree equation in the variables x and y are equations that may be put into the form Ax + By = C where A, B, C are numbers. First degree equations are the same as linear equations. They are called linear because their graphs are straight lines. To graph a linear equation, find a few ordered pairs that fit the equation.

Example B. Graph the following linear equations.

a. y = 2x – 5


First degree equation in the variables x and y are equations that may be put into the form Ax + By = C where A, B, C are numbers. First degree equations are the same as linear equations. They are called linear because their graphs are straight lines. To graph a linear equation, find a few ordered pairs that fit the equation. To find one such ordered pair, assign a value to x, plug it into the equation and solve for the y (or assign a value to y and solve for the x).


a. y = 2x – 5


First degree equation in the variables x and y are equations that may be put into the form Ax + By = C where A, B, C are numbers. First degree equations are the same as linear equations. They are called linear because their graphs are straight lines. To graph a linear equation, find a few ordered pairs that fit the equation. To find one such ordered pair, assign a value to x, plug it into the equation and solve for the y (or assign a value to y and solve for the x).


a. y = 2x – 5Make a table by selecting a few numbers for x.


First degree equation in the variables x and y are equations that may be put into the form Ax + By = C where A, B, C are numbers. First degree equations are the same as linear equations. They are called linear because their graphs are straight lines. To graph a linear equation, find a few ordered pairs that fit the equation. To find one such ordered pair, assign a value to x, plug it into the equation and solve for the y (or assign a value to y and solve for the x). For graphing lines, find at least two ordered pairs.Example B. Graph the following linear equations.

a. y = 2x – 5Make a table by selecting a few numbers for x.



a. y = 2x – 5Make a table by selecting a few numbers for x. For easy calculation let’s select x = -1, 0, 1, and 2.



a. y = 2x – 5Make a table by selecting a few numbers for x. For easy calculation let’s select x = -1, 0, 1, and 2. Plug these value into x and solve for y, one at a time, to obtain four ordered pair as shown below.


For y = 2x – 5:

x y -1 0 1 2


For y = 2x – 5:

x y -1 0 1 2

If x = -1, then y = 2(-1) – 5


For y = 2x – 5:

x y -1 -7 0 1 2

If x = -1, then y = 2(-1) – 5 = -7


For y = 2x – 5:

x y -1 -7 0 -5 1 2

If x = -1, then y = 2(-1) – 5 = -7 If x = 0, then y = 2(0) – 5


For y = 2x – 5:

x y -1 -7 0 -5 1 2

If x = -1, then y = 2(-1) – 5 = -7 If x = 0, then y = 2(0) – 5 = -5


For y = 2x – 5:

x y -1 -7 0 -5 1 -3 2 -1


If x = 1, then y = 2(1) – 5 = -3 If x = 2, then y = 2(2) – 5 = -1


For y = 2x – 5:

x y -1 -7 0 -5 1 -3 2 -1



Linear Equations and LinesPlot these ordered pairs,

For y = 2x – 5:

x y -1 -7 0 -5 1 -3 2 -1




(1,–7)

For y = 2x – 5:

x y -1 -7 0 -5 1 -3 2 -1




(1,–7)

(0,–5)

For y = 2x – 5:

x y -1 -7 0 -5 1 -3 2 -1




(1,–7)

(0,–5)

(1,–3)

(2,–1)

Plot these ordered pairs,

For y = 2x – 5:

x y -1 -7 0 -5 1 -3 2 -1




(1,–7)

(0,–5)

(1,–3)

(2,–1)

Plot these ordered pairs,then connect the dots to form the line.

b. -3y = 12 Linear Equations and Lines

b. -3y = 12 Simplify as y = -4 Make a table by selecting a few numbers for x.


b. -3y = 12 Simplify as y = -4 Make a table by selecting a few numbers for x.

x y -3 0 3 6


b. -3y = 12 Simplify as y = -4 Make a table by selecting a few numbers for x. However, y = -4 is always.

x y -3 -4 0 -4 3 -4 6 -4


b. -3y = 12 Simplify as y = -4

c. 2x = 12

Make a table by selecting a few numbers for x. However, y = -4 is always.

x y -3 -4 0 -4 3 -4 6 -4



c. 2x = 12


x y -3 -4 0 -4 3 -4 6 -4

Simplify as x = 6.



c. 2x = 12


x y -3 -4 0 -4 3 -4 6 -4

Simplify as x = 6 Make a table. However the only selction for x is x = 6



c. 2x = 12


x y -3 -4 0 -4 3 -4 6 -4

Simplify as x = 6 Make a table. However the only selction for x is x = 6

x y 6 6 6 6



c. 2x = 12


x y -3 -4 0 -4 3 -4 6 -4

Simplify as x = 6 Make a table. However the only selction for x is x = 6 and y could be any number.

x y 6 0 6 2 6 4 6 6


Summary of the graphs of linear equations:


a. y = 2x – 5



a. y = 2x – 5

If both variables x and y are present in theequation, the graph is a tilted line.



a. y = 2x – 5 b. -3y = 12




a. y = 2x – 5 b. -3y = 12


If the equation has only y (no x), the graph is a horizontal line.



a. y = 2x – 5 b. -3y = 12 c. 2x = 12





a. y = 2x – 5 b. -3y = 12 c. 2x = 12




If the equation has only x (no y), the graph is a vertical line.


The x-Intercepts is where the line crosses the x-axis;


The x-Intercepts is where the line crosses the x-axis. We set y = 0 in the equation to find the x-intercept.


The x-Intercepts is where the line crosses the x-axis. We set y = 0 in the equation to find the x-intercept. The y-Intercepts is where the line crosses the y-axis;


The x-Intercepts is where the line crosses the x-axis. We set y = 0 in the equation to find the x-intercept. The y-Intercepts is where the line crosses the y-axis. We set x = 0 in the equation to find the y-intercept.


The x-Intercepts is where the line crosses the x-axis. We set y = 0 in the equation to find the x-intercept. The y-Intercepts is where the line crosses the y-axis. We set x = 0 in the equation to find the y-intercept.Since two points determine a line, an easy method to graph linear equations is the intercept method,


The x-Intercepts is where the line crosses the x-axis. We set y = 0 in the equation to find the x-intercept. The y-Intercepts is where the line crosses the y-axis. We set x = 0 in the equation to find the y-intercept.Since two points determine a line, an easy method to graph linear equations is the intercept method, i.e. plot the x-intercept and the y intercept and the graph is the line that passes through them.



Example C. Graph 2x – 3y = 12by the intercept method.

Since two points determine a line, an easy method to graph linear equations is the intercept method, i.e. plot the x-intercept and the y intercept and the graph is the line that passes through them.


x y0

0


y-int

x-int




x y0

0


y-int

x-int



If x = 0, we get 2(0) – 3y = 12


x y0 -4

0


y-int

x-int



If x = 0, we get 2(0) – 3y = 12 so y = -4


x y0 -4

0


y-int

x-int



If x = 0, we get 2(0) – 3y = 12 so y = -4If y = 0, we get 2x – 3(0) = 12


x y0 -46 0


y-int

x-int



If x = 0, we get 2(0) – 3y = 12 so y = -4If y = 0, we get 2x – 3(0) = 12 so x = 6


Exercise. A. Solve the indicated variable for each equation with the given assigned value. 1. x + y = 3 and x = –1, find y. 2. x – y = 3 and y = –1, find x. 3. 2x = 6 and y = –1, find x. 4. –y = 3 and x = 2, find y. 5. 2y = 3 – x and x = –2 , find y. 6. y = –x + 4 and x = –4, find y. 7. 2x – 3y = 1 and y = 3, find x. 8. 2x = 6 – 2y and y = –2, find x. 9. 3y – 2 = 3x and x = 2, find y. 10. 2x + 3y = 3 and x = 0, find y. 11. 2x + 3y = 3 and y = 0, find x. 12. 3x – 4y = 12 and x = 0, find y. 13. 3x – 4y = 12 and y = 0, find x. 14. 6 = 3x – 4y and y = –3, find x.


B. a. Complete the tables for each equation with given values. b. Plot the points from the table. c. Graph the line. 15. x + y = 3 16. 2y = 6

x y -3 0 3

x y 1 0 –1

17. x = –6 x y 0 –1

– 2

18. y = x – 3 x y 2 1 0

19. 2x – y = 2 20. 3y = 6 + 2x

x y 2 0–1

x y 1 0 –1

21. y = –6

x y 0–1– 2

22. 3y + 4x =12

x y 0

0 1


C. Make a table for each equation with at least 3 ordered pairs.(remember that you get to select one entry in each row as shown in the tables above) then graph the line.23. x – y = 3 24. 2x = 6 25. –y – 7= 0

26. 0 = 8 – 2x 27. y = –x + 4 28. 2x – 3 = 6

29. 2x = 6 – 2y 30. 4y – 12 = 3x 31. 2x + 3y = 3

32. –6 = 3x – 2y 33.

35. For problems 29, 30, 31 and 32, use the intercept-tables as shown to graph the lines.

x y 0

0intercept-table

36. Why can’t we use the above intercept method to graph the lines for problems 25, 26 or 33?37. By inspection identify which equations givehorizontal lines, which give vertical lines and which give tilted lines.

3x = 4y 34. 5x + 2y = –10


57 graphing lines from linear equations

Education