Solve the following systems by substitution: x – 3y = 4 4x + 3y = –6 2x ...

[Pages:6]Solve the following systems by substitution:

a)

x ? 3y = 4 2 x + y = 3

b)

4x + 3y = ?6 5 x + y = ? 1 3

c)

xy

+ =

y x

= +

6 4

e)

2 x 6 x

? ?

3y 9y

= =

1 4

d)

23xx

= ?

?17 ? 3y

51 2

=

4y

f)

6 x 2 x

+ +

3y y

= =

?9 ?3

a)

x ? 3y = 4 2 x + y = 3

x ? 3y = 4 x = 3y + 4

2x + y = 3 2(3y + 4) + y = 3 6y + 8 + y = 3 7y = ?5 y = ?57

x = 3y + 4 = 3?57 + 4

=

?

15 7

+

4

=

?

15 7

+

28 7

=

13 7

x

=

13 7

y

=

?

7 5

b)

4x + 3y = ?6 5 x + y = ? 1 3

5x + y = ?13 y = ?5x ? 13

4x + 3y = ?6 4x + 3(?5x ? 13) = ?6 4x ? 15x ? 39 = ?6 ?11x = 33

x = ?3

y = ?5x ? 13 = ?5(?3) ? 13 = 15 ? 13 = 2

x = ?3 y= 2

c)

x + y = 6 y = x + 4

y = x + 4

x + y = 6 x + (x + 4) = 6 x + x + 4 = 6 2x = 2 x = 1

y =x+4 = (1) + 4 = 5

x = 1 y = 5

d)

23xx

= ?

?17 ?

51 2

=

3y 4y

2x = ?17 ? 3y

x

=

?

17 2

?

3 2

y

3x

?

51 2

=

4y

3?127 ?

3 2

y

?

51 2

=

4y

?

51 2

?

9 2

y

?

51 2

=

4y

?51 ? 9y ? 51 = 8y

?17y = 102

y = ?6

x

=

?

17 2

?

3 2

y

=

?

17 2

?

3 2

(?6)

=

?

17 2

+ 9

= 1/2

x

=

1 2

y = ?6

e)

2x ? 3y = 1 6x ? 9y = 4

2x ? 3y = 1

2x = 3y + 1

x

=

3 2

y

+

1 2

6x ? 9y = 4

632

y

+

1 2

?

9y

=

4

9y + 3 ? 9y = 4

0 = 1

no solution

f)

6x + 3y = ?9 2 x + y = ? 3

2x + y = ?3 y = ?2x ? 3

6x + 3y = ?9 6x + 3(?2x ? 3) = 14 6x ? 6x ? 9 = ?9

0 = 0 dependent solution

let x = c (any constant)

then y = ?2x ? 3

y = ?2c ? 3

x= c y = ?2c ? 6 c is any real number

Solve the following systems by addition:

a)

x ? 3y = 4 2 x + y = 3

b)

4x + 3y = ?6 5 x + y = ? 1 3

c)

xy

+ =

y x

= +

6 4

e)

2 x 6 x

? ?

3y 9y

= =

1 4

d)

23xx

= ?

?17 ? 3y

51 2

=

4y

f)

6 x 2 x

+ +

3y y

= =

?9 ?3

a)

x ? 3y = 4 2 x + y = 3

x ? 3y = 4 6x + 3y = 9

7x = 13

x =

13 7

x ? 3y = 4 2 x + y = 3

?22xx

+ +

6y y

= =

?8 3

7y = ?5

y

=

?

7 5

x

=

13 7

y

=

?

7 5

b)

4x + 3y = ?6 5 x + y = ? 1 3

4x + 3y = ?6 ?15x ? 3y = 39

?11x = 33

x = ?3

5x + y = ?13 5(?3) + y = ?13 ?15 + y = ?13

y =2

x = ?3 y= 2

c)

x + y = 6 y = x + 4

x + y = 6 ? x + y = 4

2y = 10 y = 5

x + y = 6 x = ?y + 6

= ?(5) + 6 = 1

x = 1 y = 5

d)

23xx

= ?

?17 ?

51 2

=

3y 4y

2x = ?17 ? 3y

6x

?

51

=

8y

2x + 3y = ?17 6 x ? 8 y = 5 1

?6x ? 9y = 51

6x

?

8y

=

51

?17y = 102 y = ?6

2x = ?17 ? 3y = ?17 ? 3(?6) = 1

x = 1/2

x

=

1 2

y = ?6

e)

2x ? 3y = 1 6x ? 9y = 4

?6x + 9y = ?3

6x

?

9y

=

4

0 =1

no solution

f)

6x + 3y = ?9 2 x + y = ? 3

6x + 3y = ?9 ? 6 x ? 3 y = 9

0 = 0 dependent solution

let x = c (any constant) then

2x + y = ?3 2c + y = ?3 y = ?2c ? 3

x= c y = ?2c ? 6 c is any real number

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