systems notecards

8/8/2019 Systems Notecards

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12.12.Using a graphing calculator to graphUsing a graphing calculator to graph

linear equationslinear equations

y = mx + by = mx + b slopeslope--interceptinterceptformform

graphgraph

table (2nd graph)table (2nd graph) xx--y tabley tablefind interceptsfind intercepts

find pointsfind points


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13. Systems of linear equations13. Systems of linear equations

Definitions:Definitions:System of linear equationsSystem of linear equations consists of 2 orconsists of 2 or

more linear equationsmore linear equations

(lines)(lines)

Solution of a system of linear equationsSolution of a system of linear equations

Ordered pair (x, y) that makes bothOrdered pair (x, y) that makes both

equations true: point of intersectionequations true: point of intersection


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14.14. Solving systems of equations bySolving systems of equations bygraphinggraphing

1.1. Solve so that both equations are inSolve so that both equations are inslopeslope--intercept form (y = mx + b)intercept form (y = mx + b)

2.2. Graph both linesGraph both lines find point offind point ofintersectionintersection

Graph on paperGraph on paper

3.3. Check your answerCheck your answer Use graphing calculator to check graph andUse graphing calculator to check graph and

tabletable


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Example 1: Tell whetherthe orderedpairisaExample 1: Tell whetherthe orderedpairisa

solution.solution.

(3, 2)(3, 2)

x + 2y = 7x + 2y = 7

3x3x 2y = 52y = 5

Example 2: Solve by graphingExample 2: Solve by graphing

y = 2x + 2y = 2x + 2y = 4x + 6y = 4x + 6


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Example 3:Example 3:

--x + y =x + y = --77

x + 4y =x + 4y = --88

Homework: p. 430Homework: p. 430 432432

#12#12--24 (124 (1stst column), #3column), #3--10 all10 all


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15.15. Solving systems of equations bySolving systems of equations by

substitutionsubstitution

We are stilllooking for the point of intersection.We are stilllooking for the point of intersection.

Steps for solving by substitutionSteps for solving by substitution

1.1. Solve for one variable in one of the equations.Solve for one variable in one of the equations.

When possible solve for the variable that has 1 orWhen possible solve for the variable that has 1 or --1 for the coefficient1 for the coefficient

2.2. Substitute the expression from step 1 into the otherSubstitute the expression from step 1 into the otherequation and solve for the remaining variable.equation and solve for the remaining variable.

3.3. Substitute the value from step 2 into the originalSubstitute the value from step 2 into the originalequation to solve for 2equation to solve for 2ndnd variable.variable.

Answer will be the point of intersection (x, y).Answer will be the point of intersection (x, y).


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Example 1:Example 1: Solve by substitutionSolve by substitutionyy == 3x + 23x + 2x + 2x + 2yy = 11= 11

Solution:Solution: Substitute expressionSubstitute expression

x + 2(x + 2(3x + 23x + 2) = 11) = 11 DistributeDistributex + 6x + 4 = 11x + 6x + 4 = 11 Combine like termsCombine like terms

7x + 4 = 117x + 4 = 11 Subtract 4 from both sidesSubtract 4 from both sides-- 44 --44

7x7x == 77 Divide both sides by 7Divide both sides by 77 77 7x = 1x = 1 Still need to find y, substituteStill need to find y, substitute-------------------------------------------------------- x into either original equationx into either original equationy = 3x + 2y = 3x + 2 and solve for yand solve for yy = 3(1) + 2y = 3(1) + 2y = 5y = 5 Answer: (1, 5) Check answerAnswer: (1, 5) Check answer


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Example 2:Example 2: xx 2y =2y = --66

4x + 6y = 4 .4x + 6y = 4 .

1. Solve 11. Solve 1

stst

equation for xequation for x xx 2y =2y = --66+2y +2y+2y +2y

x =x = 2y2y 66

2. Substitute (2y2. Substitute (2y--6) into 26) into 2ndnd

equation for x. 4(equation for x. 4(2y2y 66) + 6y = 4) + 6y = 43. Distribute3. Distribute 8y8y 24 + 6y = 424 + 6y = 4

4. Combine like terms 14y4. Combine like terms 14y 24 = 424 = 4

5. Add 24 to both sides5. Add 24 to both sides +24 +24+24 +24

14y14y == 28286. Divide both sides by 14 14 146. Divide both sides by 14 14 14

y = 2y = 2

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Continued on next slideContinued on next slide


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Only have 1 coordinate for the point of intersection, so.Only have 1 coordinate for the point of intersection, so.

7. Substitute y value into7. Substitute y value into

either original equationeither original equation 4x + 6y = 44x + 6y = 4

and solve for x 4x + 6(2) = 4and solve for x 4x + 6(2) = 4

4x + 12 = 44x + 12 = 4

--1212 --1212

4x4x == --88

44 44

x =x = --22

Answer (point of intersection): (Answer (point of intersection): (--2, 2)2, 2)Check answerCheck answer substitute into both original equations tosubstitute into both original equations to

see if true.see if true.

Homework: p. 439Homework: p. 439--440 # 3440 # 3--17 odds, 18, 31, 3217 odds, 18, 31, 32


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15. Solving systems of equations by15. Solving systems of equations by

substitutionsubstitution

Steps forsolvingSteps forsolving

1.1. Isolate onevariablein one ofthetwo equationsIsolate onevariablein one ofthetwo equations

2.2. Substitutetheexpression into the otherSubstitutetheexpression into the other

equationequation3.3. Solve forthevariable.Solve forthevariable.

4.4. Substituteanswerinto eitheroriginalequationSubstituteanswerinto eitheroriginalequation

andsolve for2andsolve for2ndnd variable.variable.


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16. Solvesystems ofequations by16. Solvesystems ofequations by

eliminationelimination

We are still looking for the point of intersection.We are still looking for the point of intersection.

Steps forsolving by eliminationSteps forsolving by elimination

1. Both equations needto bein standard form1. Both equations needto bein standard form sosothatvariableslineup.thatvariableslineup. (Ax + By = C)(Ax + By = C)

2. Oneset ofvariables needto havethesame2. Oneset ofvariables needto havethesame

coefficient but oppositesigns.coefficient but oppositesigns.

3. Addthetwo equationstogether.3. Addthetwo equationstogether.4. Solve fortheremaining variable.4. Solve fortheremaining variable.

5. Substituteanswerinto eitheroriginalequation and5. Substituteanswerinto eitheroriginalequation and

solve

for

2solve

for

2

ndnd variable

. Answer

: (x, y)

variable

. Answer

: (x, y)


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Example 1:Example 1: 2x + 3y = 112x + 3y = 11 Both Standard?Both Standard?

--2x + 5y = 132x + 5y = 13 Coefficients?Coefficients?

AddequationsAddequations8y8y == 2424 Solve forvariableSolve forvariable

8 88 8

y = 3y = 3 Still needto find 2Still needto find 2ndnd

variablevariable

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2x + 3(3) = 112x + 3(3) = 11

2x + 9 = 112x + 9 = 11-- 99 --99

2x2x == 22

2 22 2

x = 1x = 1 Answer:Answer: (1, 3)(1, 3)


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Example 2: 4x + 3y = 2Example 2: 4x + 3y = 2 Standard?Standard?

5x + 3y =5x + 3y = --22 Coefficients?Coefficients?

4x + 3y = 24x + 3y = 2 4x + 3y = 24x + 3y = 2

--1(5x + 3y =1(5x + 3y = --2)2) --5x5x 3y = 23y = 2 Now addNow add

--1x1x == 44

--11 --11

x =x = --44

Find 2Find 2ndnd variable: 4(variable: 4(--4) + 3y = 24) + 3y = 2

--16 + 3y = 216 + 3y = 216 1616 16

3y3y == 1818

3 33 3

y = 6y = 6 Answer: (Answer: (--4, 6)4, 6)


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Example 3:Example 3:

8x8x 4y =4y = --44 8x8x 4y =4y = --44

4y = 3x + 144y = 3x + 14--3x3x --3x3x --3x + 4y = 143x + 4y = 14

5x5x == 1010

5 55 5

x = 2x = 2

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8(2)8(2) 4y =4y = --44

1616 4y =4y = --44--1616 --1616

--4y4y == --2020

--44 --44

y = 5y = 5 Answer: (2, 5)Answer: (2, 5)


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Homework: p. 447Homework: p. 447-- 448448 #3#3 21 odds, 2521 odds, 25


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16. Solving systems by elimination16. Solving systems by elimination

1. Both equations need to be in standard form1. Both equations need to be in standard form

2. One set of coefficients must have the same number but2. One set of coefficients must have the same number but

opposite signs. Multiply one or both equationsopposite signs. Multiply one or both equations

3. Add the equations together3. Add the equations together4. Solve for the remaining variable4. Solve for the remaining variable

5. Substitute value for variable into one of the two original5. Substitute value for variable into one of the two original

equations and solve for the 2equations and solve for the 2ndndvariablevariable


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Example 1:Example 1: 6x + 5y = 196x + 5y = 19

2x + 3y = 52x + 3y = 5

6x + 5y = 196x + 5y = 19 6x + 5y = 196x + 5y = 19

--3(2x + 3y = 5)3(2x + 3y = 5) --6x6x 9y =9y = --1515 AddAdd

-- 4y4y == 44

--44 --44

y =y = --11

2x + 3(2x + 3(--1) = 51) = 5

2x2x -- 3 = 53 = 5

3 33 3

2x2x == 88 x = 4x = 4 (4,(4, --1)1)

2 22 2


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--3x + 2y =3x + 2y = --99

3(4x + 5y = 35)3(4x + 5y = 35) 12x + 15y = 10512x + 15y = 105

4(4(--3x + 2y =3x + 2y = --9)9) --12x + 8y =12x + 8y = --3636

23y23y == 6969

23 2323 23y = 3y = 3

4x + 5(3) = 354x + 5(3) = 35

4x + 15 = 354x + 15 = 35

--1515 --1515

4x4x == 2020

44 44

x = 5x = 5 Answer: (5, 3)Answer: (5, 3)


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Example 3:Example 3: 6x6x 2y = 12y = 1

--2x + 3y =2x + 3y = --55


3x + 10y =3x + 10y = --33

Example 5:Example 5: 3x3x 7y = 57y = 5

9y = 5x + 59y = 5x + 5

Homework: p. 454Homework: p. 454 455455 #3#3--17 odds, 21, 24, 27, 3717 odds, 21, 24, 27, 37


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What is a system of linear equations?What is a solution for a system of equations?

Is a single point of intersection the only possiblilitywhen graphing two lines?

Graph the only possibility when y = 2x 3 and y =2x +5.

When graphing two lines it is easy to see what ishappening with the two lines. But whathappens when we just have two equations, howare we going to figure out when we have 1 pointof intersection, parallel lines (no solution) or the

same line (infinite solutions)?


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17. Special types of linear systems

1. Solve using substitution or elimination

2. Our answers will tell us what ishappening.

If we get x = or y = , we are looking for a

single point of intersection If we get a false statement without a

variable (0 = 5), then we have parallel linesand no common points. The answer is no

solution. If we get a true statement without avariable (-3 = -3), then we have the sameline and all the points are in common. Theanswer is infinite solutions.


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Example 1: 3x + 2y = 10

3x + 2y = 2

Use elimination to solve.

3x + 2y = 10 3x + 2y = 10

-1(3x + 2y = 2) -3x 2y = -2 Add

0 = 8

False parallel lines - no solutions


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Example 2: x 2y = -4

y = x + 2

Use substitution.

x 2(x + 2) = -4x x 4 = -4

-4 = -4

True same line infinite solutions


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Homework: p.462-463 #5-7 all, 9-13 odds, 15, 18,21

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