Today:
Grades & Weighting:
All Quarters
# of Assignments

Weight

Class Work

10 Graded

20%

Home Work
Tests
Notebook

6
6
1

20...
5 Most Missed Questions from Final Exam
#5; Version 1: 39% correct

Mrs. Wright rents a car to see her friend Mrs. Wong. T...
5 Most Missed Questions from Final Exam
#3; Version 2: 36% correct

A. 125%

B. 90%

C. 55%

D. 80%

E. None
5 Most Missed Questions from Final Exam
#2; Version 3: 29% correct

Bob made a fresh pot of coffee in the morning. By 10:0...
SYSTEMS OF LINEAR EQUATIONS
So far, we have solved equations with one variable: 3x + 5 =
35
and in two variables. 3x + 5y ...
WHAT IS A SYSTEM OF LINEAR EQUATIONS?
If the system of linear equations is going to have a solution,
then the solution wil...
Tell whether the ordered pair is a solution of
the given system.
(5, 2);
3x – y = 13

Substitute 5 for x and 2
for y in ea...
Helpful Hint
If an ordered pair does not satisfy the first equation in the
system, there is no reason to check the other e...
SOLVING LINEAR SYSTEMS BY GRAPHING
Consider the following system:

Using the graph to the right,
we can see that any of th...
Practice 1
Graph the system of equations. Determine whether the
system has one solution, no solution, or infinitely many
s...
Practice 1
y

The two equations in slopeintercept form are:

x

Plot points for each line.
Draw the lines.
These two equat...
Practice 2
y
The two equations in
slope-intercept form are:

x

Plot points for each line.
Draw in the lines.

This system...
Practice 3
The two equations in
slope-intercept form are:

y

x
Plot points for each line.

Draw in the lines.

This syste...
Graphing to Solve a Linear System
Let's summarize! There are 4 steps to solving a linear
system using a graph.
Step 1: Put...
Solve: by ELIMINATION

5x - 4y = -21
-2x + 4y = 18

3x + 0 = -3

x = -1

Like variables must be lined under each other.

W...
5x - 4y = -21
5(-1) – 4y = -21
Substitute your first
solution into either original -5 – 4y = -21
equation and solve for th...
5X - 4Y = -21
5(-1) – 4(4) = -21
-5 - 16 = -21
-21 = -21
-2x + 4y = 18
-2(-1) + 4(4) = 18
2 + 16 = 18
18 = 18
Solve:
by ELIMINATION
x + y = 30
x + 7y = 6
We need to eliminate (get rid of) a variable.
To simply add this time will not...
X + Y = 30
( X + 7Y = 6) -1
Now add the two
equations and solve.

-X – 7Y = - 6
-6Y = 24
-6

-6

Y=-4
THEN----
Substitute
your answer
into either
original
equation and
solve for the
second
variable.

X + Y = 30
X + - 4 =30
4 4
X = 34...
x + y = 30
34 + - 4 = 30
30 = 30

x + 7y = 6
34 + 7(- 4) = 6
34 - 28 = 6
6=6
Solve: Elimination By
Multiplying
Like variables
must be lined
under each other.

x + +y0y 4 4
0x = =
2x + 3y = 9

We need...
(X + Y = 4 ) -2

-2X - 2 Y = - 8

2X + 3Y = 9

2X + 3Y = 9

Now add the two
equations and solve.

THEN----

Y=1
Substitute your answer into either original
equation and solve for the second variable.

X+Y=4

X +1=4
- 1 -1
X=3
Solution...
x+y=4
3+1=4
4=4

2x + 3y = 9
2(3) + 3(1) = 9
6+3=9

9=9
Class Work
Many states fine speeders $15 for each mile per hour over
the speed limit of 45 mph. Graph this relationship
a) Determine ...
Fine (cost)

e) What is the fine
for a driver going
68 MPH? $345

y = 15(x – 45)

250

f) Is this a
function? No
g) Is thi...
Odds n’ Ends
REVIEW: SOLVING BY GRAPHING
If the lines cross once, there will be one solution.
If the lines are parallel, there will be ...
Writing Equations of Lines
C. Given a point and the equation of a line parallel to it.

Find the equation of the line that...
Writing Equations of Lines
D. Given a point and equation of a line perpendicular to
it.
Find the equation of the line that...
Solve:
by ELIMINATION
2x + 7y = 31
5x - 7y = - 45
x = -2
7x + 0 = -14
Like variables must be lined under each other.

THEN...
Substitute
your answer
into either
original
equation and
solve for the
second
variable.

2X + 7Y = 31
2(-2) + 7y = 31
-4 +...
2x + 7y = 31
2(-2) + 7(5) = 31
-4 + 35 = 31
31 = 31

5x – 7y = - 45
5(-2) - 7(5) = - 45

-10 - 35 = - 45
- 45 =- 45
Solve: By Substitution
Recall that when we 'solve' a point-slope formula, we
end up in slope-intercept form. In much the s...
Solve: By Substitution
Example 1: y = 2x
4x - y = -4
Example 1: Substitute 2x for y in the 2nd equation
y = 2x
4x - 2x = -...
Solve: By Substitution
Example 2: 3x + 5y = -7
x = 2y + 5

Example 3: y = 2x - 1
6x - 3y = 7
Applying Systems of Equations
Solve by Elimination.
Example 1: The sum of two numbers is 52. The
larger number is 2 more t...
3x - 2y = 8
2x - 2y = 4

3x - 2y = 8
-6x - 5y = -12

3x - 2y = 8
4x + y = 2
Class Work:
January 29, 2014
January 29, 2014
January 29, 2014
January 29, 2014
January 29, 2014
January 29, 2014
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January 29, 2014

  1. 1. Today:
  2. 2. Grades & Weighting: All Quarters # of Assignments Weight Class Work 10 Graded 20% Home Work Tests Notebook 6 6 1 20% 40% 10% Final Exam 1 10%
  3. 3. 5 Most Missed Questions from Final Exam #5; Version 1: 39% correct Mrs. Wright rents a car to see her friend Mrs. Wong. The cost is $32.00 plus $.75 a mile. The trip is 45 miles each way. What is the total cost for Wright to see Wong? A. $32.75 B. $99.50 C. $75.00 D. $65.75 E. None #4; Version 1: 37% correct Solve for p: A = (ap + pH) A) p = a + H A B) p = A C) p = A – a D) p = H - a a+H H A E. None
  4. 4. 5 Most Missed Questions from Final Exam #3; Version 2: 36% correct A. 125% B. 90% C. 55% D. 80% E. None
  5. 5. 5 Most Missed Questions from Final Exam #2; Version 3: 29% correct Bob made a fresh pot of coffee in the morning. By 10:00 am only 3 cups remained. If 90% of the coffee had been consumed, how many cups of coffee did Bob make in total? A. 22 B. 15 #1; Version 3: 26% correct C. 18 D. 30 E. 20
  6. 6. SYSTEMS OF LINEAR EQUATIONS So far, we have solved equations with one variable: 3x + 5 = 35 and in two variables. 3x + 5y = That will change now as 35 In both cases, though we solve multiple we have only been able equations at the same to solve one equation at time, looking for an a time. ordered pair which solves each equation, and thus is a solution for both. Example: There are 3 methods for solving systems of 3x + 3y = -3 equations: 1) by Graphing 2) By Elimination y=x+1 3) By Substitution Using any of the 3 systems will show the solution to these equations is x = -1, y = 0
  7. 7. WHAT IS A SYSTEM OF LINEAR EQUATIONS? If the system of linear equations is going to have a solution, then the solution will be an ordered pair (x , y) where x and y make both equations true at the same time. If the lines are parallel, there will be no solutions. If the equations are the same line, there will be an infinite number of solutions. All other lines will have a single solution. There are several methods of solving systems of equations; we'll look at a couple today.
  8. 8. Tell whether the ordered pair is a solution of the given system. (5, 2); 3x – y = 13 Substitute 5 for x and 2 for y in each equation in the system. 3x – y 13 0 3(5) – 2 13 2–2 0 0 0 15 – 2 13 13 13 The ordered pair (5, 2) makes both equations true. (5, 2) is the solution of the system.
  9. 9. Helpful Hint If an ordered pair does not satisfy the first equation in the system, there is no reason to check the other equations. x + 3y = 4 (–2, 2); –x + y = 2 x + 3y = 4 –2 + 3(2) 4 –2 + 6 4 4 4 Substitute –2 for x and 2 for y in each equation in the system. –x + y = 2 – (–2) + 2 2 4 2 The ordered pair (–2, 2) makes one equation true but not the other. (–2, 2) is not a solution of the system.
  10. 10. SOLVING LINEAR SYSTEMS BY GRAPHING Consider the following system: Using the graph to the right, we can see that any of these ordered pairs will make the first equation true since they lie on the line. Notice that any of these points will make the second equation true. x – y = –1 x + 2y = 5 y (1 , 2) However, there is ONE point that makes both true at the same time… The point where they intersect makes both equations true at the same time. x
  11. 11. Practice 1 Graph the system of equations. Determine whether the system has one solution, no solution, or infinitely many solutions. If the system has one solution, determine the solution.
  12. 12. Practice 1 y The two equations in slopeintercept form are: x Plot points for each line. Draw the lines. These two equations represent the same line. Therefore, this system of equations has infinitely many solutions .
  13. 13. Practice 2 y The two equations in slope-intercept form are: x Plot points for each line. Draw in the lines. This system of equations represents two parallel lines. This system of equations has no solution because these two lines have no points in common.
  14. 14. Practice 3 The two equations in slope-intercept form are: y x Plot points for each line. Draw in the lines. This system of equations represents two intersecting lines. The solution to this system of equations is a single point (3,0)
  15. 15. Graphing to Solve a Linear System Let's summarize! There are 4 steps to solving a linear system using a graph. Step 1: Put both equations in slope - intercept form. Solve both equations for y, so that each equation looks like y = mx + b. Step 2: Graph both equations on the same coordinate plane. Use the slope and y - intercept for each equation in step 1. Step 3: Plot the point where the graphs intersect. This is the solution! LABEL the solution! Step 4: Check to make sure your solution makes both equations true. Substitute the x and y values into both equations to verify the point is a solution to both equations.
  16. 16. Solve: by ELIMINATION 5x - 4y = -21 -2x + 4y = 18 3x + 0 = -3 x = -1 Like variables must be lined under each other. We need to eliminate (get rid of) a variable by cancelling out one of the variables. We then solve for the other variable. The y’s be will the easiest. So, we will add the two equations. THEN----
  17. 17. 5x - 4y = -21 5(-1) – 4y = -21 Substitute your first solution into either original -5 – 4y = -21 equation and solve for the 5 5 second variable. -4y = -16 y = 4 The solution to this system of equations is: (-1, 4) Now check your answers in both equations------
  18. 18. 5X - 4Y = -21 5(-1) – 4(4) = -21 -5 - 16 = -21 -21 = -21 -2x + 4y = 18 -2(-1) + 4(4) = 18 2 + 16 = 18 18 = 18
  19. 19. Solve: by ELIMINATION x + y = 30 x + 7y = 6 We need to eliminate (get rid of) a variable. To simply add this time will not eliminate a variable. If one of the x’s was negative, it would be eliminated when we add. So we will multiply one equation by a – 1.
  20. 20. X + Y = 30 ( X + 7Y = 6) -1 Now add the two equations and solve. -X – 7Y = - 6 -6Y = 24 -6 -6 Y=-4 THEN----
  21. 21. Substitute your answer into either original equation and solve for the second variable. X + Y = 30 X + - 4 =30 4 4 X = 34 Solution (34, - 4) Now check our answers in both equations------
  22. 22. x + y = 30 34 + - 4 = 30 30 = 30 x + 7y = 6 34 + 7(- 4) = 6 34 - 28 = 6 6=6
  23. 23. Solve: Elimination By Multiplying Like variables must be lined under each other. x + +y0y 4 4 0x = = 2x + 3y = 9 We need to eliminate (get rid of) a variable. To simply add this time will not eliminate a variable. If there was a – 2x in the 1st equation, the x’s would be eliminated when we add. So we will multiply the 1st equation by a – 2.
  24. 24. (X + Y = 4 ) -2 -2X - 2 Y = - 8 2X + 3Y = 9 2X + 3Y = 9 Now add the two equations and solve. THEN---- Y=1
  25. 25. Substitute your answer into either original equation and solve for the second variable. X+Y=4 X +1=4 - 1 -1 X=3 Solution (3,1) Now check our answers in both equations--
  26. 26. x+y=4 3+1=4 4=4 2x + 3y = 9 2(3) + 3(1) = 9 6+3=9 9=9
  27. 27. Class Work
  28. 28. Many states fine speeders $15 for each mile per hour over the speed limit of 45 mph. Graph this relationship a) Determine the dependent & independent variables b) Write the equation representing fines as a relation to speed. Test your equation with real numbers to see if it fits. (Your first equation may not be the correct one, so try others. c) Label the axes, plot the x-intercept and at least one other point. d) What is the speed of the driver given a $180 ticket for driving over the 45 MPH speed limit? e) What is the fine for a driver going 68 MPH? f) Is this a function? g) Is this a linear equation?
  29. 29. Fine (cost) e) What is the fine for a driver going 68 MPH? $345 y = 15(x – 45) 250 f) Is this a function? No g) Is this a linear equation? No (60,225) 150 50 20 (45,0) Speed
  30. 30. Odds n’ Ends
  31. 31. REVIEW: SOLVING BY GRAPHING If the lines cross once, there will be one solution. If the lines are parallel, there will be no solutions. If the lines are the same, there will be an infinite number of solutions.
  32. 32. Writing Equations of Lines C. Given a point and the equation of a line parallel to it. Find the equation of the line that passes throug is parallel to 4x – 2y =3. Rewrite the equation to slopeintercept form to get the slope. Solution: y1 = -5x1 = 1
  33. 33. Writing Equations of Lines D. Given a point and equation of a line perpendicular to it. Find the equation of the line that passes through (1, -5) and is perpendicular to 4x – 2y =3. Solution: x1 = 1 y1 = -5 Rewrite the equation to slope- intercept form to get the slope. m= -
  34. 34. Solve: by ELIMINATION 2x + 7y = 31 5x - 7y = - 45 x = -2 7x + 0 = -14 Like variables must be lined under each other. THEN----
  35. 35. Substitute your answer into either original equation and solve for the second variable. 2X + 7Y = 31 2(-2) + 7y = 31 -4 + 7y = 31 4 4 7y = 35 y=5 (-2, 5) Now check our answers in both equations-----Answer
  36. 36. 2x + 7y = 31 2(-2) + 7(5) = 31 -4 + 35 = 31 31 = 31 5x – 7y = - 45 5(-2) - 7(5) = - 45 -10 - 35 = - 45 - 45 =- 45
  37. 37. Solve: By Substitution Recall that when we 'solve' a point-slope formula, we end up in slope-intercept form. In much the same way, the substitution method is closely related to the elimination method. After eliminating one variable and solving for the other, we substitute the value of the variable back into the equation. For example: Solve 2x + 3y = -26 using elimination What is the value of x ? 4x - 3y = 2 At this point we substitute -4 for x, and solve for y. This is exactly what the substitution method is except it is done at the beginning.
  38. 38. Solve: By Substitution Example 1: y = 2x 4x - y = -4 Example 1: Substitute 2x for y in the 2nd equation y = 2x 4x - 2x = -4; 2x = -4; x = -2 Then, substitute -2 for x in the first equation: y = 2(-2); y = -4 Finally, plug both values in and check for equality. -4 = 2(-2); True; 4(-2) - (-4) = -4; -8 + 4 = -4; True
  39. 39. Solve: By Substitution Example 2: 3x + 5y = -7 x = 2y + 5 Example 3: y = 2x - 1 6x - 3y = 7
  40. 40. Applying Systems of Equations Solve by Elimination. Example 1: The sum of two numbers is 52. The larger number is 2 more than 4 times the smaller number. Find both numbers. Example 1: -(x ++y ==-52 -x - y 52) x 52 x = 4y=+2 -4y + _________2 Rearrange -5y = -50 y = 10 x + 10 = 52; x = 42
  41. 41. 3x - 2y = 8 2x - 2y = 4 3x - 2y = 8 -6x - 5y = -12 3x - 2y = 8 4x + y = 2
  42. 42. Class Work:

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