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Today:
 Warm-Up: (3)
 Review Systems Solutions: (6)
 Solving 3x3 Systems:(2)
 Complete Class Work from Yesterday
Warm Up
1. Write an equation for a line
perpendicular to 2x -4y = -2
2. Solve for a: 9a – 2b = c + 4a
a. Write the equation of the line
b. Write the inequality of the graph.
4. Write the systems of equations
shown by the graph below.
Graphing to Solve a Linear System
Step 1: Put both equations in
slope - intercept form.
Step 2: Graph both equations on
the same coordinate plane.
Step 3: Plot the point where the
graphs intersect.
Step 4: Check to make sure your
solution makes both equations true.
Solve both equations for y, so that
each equation looks like
y = mx + b.
Use the slope and y - intercept
for each equation in step 1.
This is the solution! LABEL the
solution!
Substitute the x and y values into
both equations to verify the point is
a solution to both equations.
4-Step Summary
Review: Solve Systems of Equations by Graphing: (1)
( + 0)
Step 1: Put both equations in
slope - intercept form.
Step 2: Graph both equations on
the same coordinate plane.
Step 3: Plot the point where the
graphs intersect.
(2,1)
Step 4: Check to make sure your
solution makes both equations true.
1 = 1 + 0
2 + 1 = 3
1: Put the equations in
Standard Form.
Step 2: Determine which
variable to eliminate.
Step 3: Add or subtract
the equations.
Step 4: Plug back in to
find other variable.
Step 5: Check your
solution.
Standard Form: Ax + By = C
Look for variables that have
the same coefficient.
Solve for the variable.
Substitute the value of the
variable into the equation.
Substitute your ordered pair
into BOTH equations.
Elimination is easiest when the equations are in
standard form.
Solve Systems of Equations by Elimination
(add. or subtract)
2x + 7y = 31
5x - 7y = - 45
7x + 0 = -14 x = -2
THEN----
Like variables must be lined under each other.
Solve Systems of Equations by Elimination:(2)
(addition or subtraction)
2x + 7y = 31
(-2, 5)
Substitute your
answer into either
original equation
and solve for the
second variable.
Solution
Now check our answers in both equations------
2(-2) + 7y = 31
-4 + 7y = 31
4 4
7y = 35; y = 5
Review: Solve Systems of Equations by Elimination
(addition or subtraction)
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
What variable to eliminate?
Like variables
must be lined
under each
other.
x + y = 4
2x + 3y = 9
1x + 1y = 4
2x + 3y = 9
Solve Systems of Equations by Elimination
(multiplying)
By doing what?
2x + 3y = 9
-2x - 2 y = - 8
2x + 3y = 9
Now add the two
equations and solve.
y = 1
THEN----
x + y = 4( ) -2
Solve Systems of Equations by Elimination
(multiplying)
(3,1)
Substitute your
answer into either
original equation
and solve for the
second variable.
Solution
Now check our answers in both equations--
x + y = 4
x + 1 = 4
- 1 -1
x = 3
Solve Systems of Equations by Elimination
(multiplying)
x + y = 4
3 + 1 = 4
4 = 4
2x + 3y = 9
2(3) + 3(1) = 9
6 + 3 = 9
9 = 9
Can you multiply either equation
by an integer in order to eliminate
one of the variables?
Here, we must multiply both
equations by a (different)
number in order to easily
eliminate one of the variables.
Multiply the top
equation by 2, and the
bottom equation by -3
 Eliminate
 Plug back in
 Solve for other
variable
3x – 2y = -7
2x -5y = 10
Write your solution as
an ordered pair
(-5,-4)
Plug both solutions into
original equations
Solve Systems of Equations by Elimination
(multiplying)
3x – 2y = -7
-15 – (-8) = -7
-7 = - 7
2x - 5y = 10
-10 – (-20) = 10
10= 10
Solving a system of equations by substitution
Step 1: Solve an equation for
one variable.
Step 2: Substitute
Step 3: Solve the equation.
Step 4: Plug back in to find
the other variable.
Step 5: Check your
solution.
Pick the easier equation. The goal
is to get y= ; x= ; a= ; etc.
Put the equation solved in Step 1
into the other equation.
Get the variable by itself.
Substitute the value of the variable
into the equation.
Substitute your ordered pair into
BOTH equations.
Solve: By Substitution: (2)
The same way, the substitution method is very
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.
2x + 3y = -26
4x - 3y = 2
What is the
value of x ?
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.
-4
For example: Find the value of ‘x’ using elimination.
Solve By Substitution: (2)
Example 1: 2x + y = 0
4x - y = -4
Solve for y in the first equation
Example 2: y = 2x - 1
6x - 3y = 7
Substitution won’t always
be so obvious. You may
have to rearrange the
equation to isolate the
variable.
Solving 3x3 Systems:(2)
The graph of the solution set of an equation in three
variables is a plane, not a line. In fact, graphing equations in
three variables requires the use of a three-dimensional
coordinate system. It is therefore, not practical to solve
3x3 systems by graphing
Solve the System: 4x + 2y - z = -5
3y + z = -1
2z = 10
1. Which is the easiest
variable to solve for?
2. Plug in where and solve
for what?
3. Substitute y and z values into equation 1; solve for x.
4. Substitute all values, check for equality.
5. The solution set is (1, -2, 5)
Solving 3x3 Systems: (2)
Practice 2:
-3x - 2y - z = -17
3x + 5y = 29
-5x = -15
Solving 3x3 Systems: (2)
5. A restaurant charged one customer $28.20 for 3 small dishes and 5 large dishes
and charged another customer $23.30 for 4 small dishes and 3 large dishes. What
will 2 small and 4 large dishes cost?
A shopper bought two pairs of gloves and four hats
for $42.00. Her friend bought two pairs of gloves and
two hats for $30.00. What is the price of each item?
Use ‘g’ for gloves and ‘h’ for hats.
Equation 1: 2g + 4h = $42.00
Equation 2: 2g + 2h = $30.00
0g + 2h = $12.00
Hats are $6.00 each
2g + 24 = $42.00
Plug both values in
and check for
equality.
Systems of Equations Word Problems: (2)
Gloves are $9.00 ea.
For Valentines Day Mark bought his mom 12 flowers, a
mixture of roses and daisies. The roses cost $1.15 each
and the daisies cost $1.35 each. If he spent $16.00, how
many daisies did he buy?
x =
y =
# of
roses
# of
daisies
12x y 
1.15 1.35 16.00x y 
 100  100  100
115x 135y 1600
115x
12x y 
  12x  11
11 11 
1x 
11 daisies
1. Mark
the text.
2. Label
variables.
3. Create
equations.
4. Solve.
5. Check.
 115  115  115
115y 1380 
20 20
11y 
20 220y 
Let’s eliminate
the ‘x’
Try solving by elimination
Complete all class work
problems

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February 13, 2015

  • 1. Today:  Warm-Up: (3)  Review Systems Solutions: (6)  Solving 3x3 Systems:(2)  Complete Class Work from Yesterday
  • 2. Warm Up 1. Write an equation for a line perpendicular to 2x -4y = -2 2. Solve for a: 9a – 2b = c + 4a a. Write the equation of the line b. Write the inequality of the graph.
  • 3. 4. Write the systems of equations shown by the graph below.
  • 4.
  • 5. Graphing to Solve a Linear System Step 1: Put both equations in slope - intercept form. Step 2: Graph both equations on the same coordinate plane. Step 3: Plot the point where the graphs intersect. Step 4: Check to make sure your solution makes both equations true. Solve both equations for y, so that each equation looks like y = mx + b. Use the slope and y - intercept for each equation in step 1. This is the solution! LABEL the solution! Substitute the x and y values into both equations to verify the point is a solution to both equations. 4-Step Summary
  • 6. Review: Solve Systems of Equations by Graphing: (1) ( + 0) Step 1: Put both equations in slope - intercept form. Step 2: Graph both equations on the same coordinate plane. Step 3: Plot the point where the graphs intersect. (2,1) Step 4: Check to make sure your solution makes both equations true. 1 = 1 + 0 2 + 1 = 3
  • 7. 1: Put the equations in Standard Form. Step 2: Determine which variable to eliminate. Step 3: Add or subtract the equations. Step 4: Plug back in to find other variable. Step 5: Check your solution. Standard Form: Ax + By = C Look for variables that have the same coefficient. Solve for the variable. Substitute the value of the variable into the equation. Substitute your ordered pair into BOTH equations. Elimination is easiest when the equations are in standard form. Solve Systems of Equations by Elimination (add. or subtract)
  • 8. 2x + 7y = 31 5x - 7y = - 45 7x + 0 = -14 x = -2 THEN---- Like variables must be lined under each other. Solve Systems of Equations by Elimination:(2) (addition or subtraction)
  • 9. 2x + 7y = 31 (-2, 5) Substitute your answer into either original equation and solve for the second variable. Solution Now check our answers in both equations------ 2(-2) + 7y = 31 -4 + 7y = 31 4 4 7y = 35; y = 5 Review: Solve Systems of Equations by Elimination (addition or subtraction)
  • 10. 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
  • 11. What variable to eliminate? Like variables must be lined under each other. x + y = 4 2x + 3y = 9 1x + 1y = 4 2x + 3y = 9 Solve Systems of Equations by Elimination (multiplying) By doing what?
  • 12. 2x + 3y = 9 -2x - 2 y = - 8 2x + 3y = 9 Now add the two equations and solve. y = 1 THEN---- x + y = 4( ) -2 Solve Systems of Equations by Elimination (multiplying)
  • 13. (3,1) Substitute your answer into either original equation and solve for the second variable. Solution Now check our answers in both equations-- x + y = 4 x + 1 = 4 - 1 -1 x = 3 Solve Systems of Equations by Elimination (multiplying)
  • 14. x + y = 4 3 + 1 = 4 4 = 4 2x + 3y = 9 2(3) + 3(1) = 9 6 + 3 = 9 9 = 9
  • 15. Can you multiply either equation by an integer in order to eliminate one of the variables? Here, we must multiply both equations by a (different) number in order to easily eliminate one of the variables. Multiply the top equation by 2, and the bottom equation by -3  Eliminate  Plug back in  Solve for other variable 3x – 2y = -7 2x -5y = 10 Write your solution as an ordered pair (-5,-4) Plug both solutions into original equations Solve Systems of Equations by Elimination (multiplying)
  • 16. 3x – 2y = -7 -15 – (-8) = -7 -7 = - 7 2x - 5y = 10 -10 – (-20) = 10 10= 10
  • 17. Solving a system of equations by substitution Step 1: Solve an equation for one variable. Step 2: Substitute Step 3: Solve the equation. Step 4: Plug back in to find the other variable. Step 5: Check your solution. Pick the easier equation. The goal is to get y= ; x= ; a= ; etc. Put the equation solved in Step 1 into the other equation. Get the variable by itself. Substitute the value of the variable into the equation. Substitute your ordered pair into BOTH equations.
  • 18. Solve: By Substitution: (2) The same way, the substitution method is very 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. 2x + 3y = -26 4x - 3y = 2 What is the value of x ? 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. -4 For example: Find the value of ‘x’ using elimination.
  • 19. Solve By Substitution: (2) Example 1: 2x + y = 0 4x - y = -4 Solve for y in the first equation Example 2: y = 2x - 1 6x - 3y = 7 Substitution won’t always be so obvious. You may have to rearrange the equation to isolate the variable.
  • 20. Solving 3x3 Systems:(2) The graph of the solution set of an equation in three variables is a plane, not a line. In fact, graphing equations in three variables requires the use of a three-dimensional coordinate system. It is therefore, not practical to solve 3x3 systems by graphing Solve the System: 4x + 2y - z = -5 3y + z = -1 2z = 10 1. Which is the easiest variable to solve for? 2. Plug in where and solve for what? 3. Substitute y and z values into equation 1; solve for x. 4. Substitute all values, check for equality. 5. The solution set is (1, -2, 5) Solving 3x3 Systems: (2)
  • 21. Practice 2: -3x - 2y - z = -17 3x + 5y = 29 -5x = -15 Solving 3x3 Systems: (2) 5. A restaurant charged one customer $28.20 for 3 small dishes and 5 large dishes and charged another customer $23.30 for 4 small dishes and 3 large dishes. What will 2 small and 4 large dishes cost?
  • 22. A shopper bought two pairs of gloves and four hats for $42.00. Her friend bought two pairs of gloves and two hats for $30.00. What is the price of each item? Use ‘g’ for gloves and ‘h’ for hats. Equation 1: 2g + 4h = $42.00 Equation 2: 2g + 2h = $30.00 0g + 2h = $12.00 Hats are $6.00 each 2g + 24 = $42.00 Plug both values in and check for equality. Systems of Equations Word Problems: (2) Gloves are $9.00 ea.
  • 23. For Valentines Day Mark bought his mom 12 flowers, a mixture of roses and daisies. The roses cost $1.15 each and the daisies cost $1.35 each. If he spent $16.00, how many daisies did he buy? x = y = # of roses # of daisies 12x y  1.15 1.35 16.00x y   100  100  100 115x 135y 1600 115x 12x y    12x  11 11 11  1x  11 daisies 1. Mark the text. 2. Label variables. 3. Create equations. 4. Solve. 5. Check.  115  115  115 115y 1380  20 20 11y  20 220y  Let’s eliminate the ‘x’ Try solving by elimination
  • 24. Complete all class work problems