Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ π -¹ ² ³ °

You are not logged in.

- Topics: Active | Unanswered

Pages: **1**

**demha****Member**- Registered: 2012-11-25
- Posts: 195

This is for my school work. I am working online. Before I send it in I would like to know if it's all correct.

Solving Systems of Equations Using Substitution Method and Elimination Method:

SUBSTITUTION METHOD

1.

y = (2/3)x - 1

y = -x + 4

y = (2/3) - 1

(-x + 4) = (2/3)x 1

(-1 1/2)x + 4 = -1

(-1 1/2)x = -5

x = 3

y = -x + 4

y = -3 + 4

y = 1

Answer: (3, 1)

2.

x + y = 0

3x + y = -4

x + y = 0

-x -x

y = 0 x

3x + (0 x) = -4

2x + 0 = -4

2x = -4

x = -2

x + y = 0

(-2) + y = 0

+2 +2

y = 0 + 2

y = 2

Answer: (-2, 2)

3.

4x + 3y = -15

y = x + 2

4x + 3(x + 2) = -15

4x + (3x + 6) = -15

7x + 6 = -15

-6 -6

7x = -21

x = -3

y = x + 2

y = (-3) + 2

y = -1

Answer: (-3, -1)

4.

x + 2y = -4

4y = 3x + 12

x + 2y = -4

x = -4 - 2y

4y = 3(-4 - 2y) + 12

4y = (-12 6y) + 12

10y = -12 + 12

10y = 0

y = 0

4y = 3x + 12

4(0) = 3x + 12

3x = 12

x = 4

Answer: (4, 0)

5.

y = 2x

x + y = 3

x + y = 3

x + (2x) = 3

3x = 3

x = 1

y = 2x

y = 2(1)

y = 2

Answer: (1, 2)

6.

x = 3 - 3y

x + 3y = -6

x + 3y = -6

(3 3y) + 3y = -6

3 + 0 = -6

3 = -6

Answer: There is no solution.

7.

y = -2x + 1

y = x 5

y = -2x + 1

(x 5) = -2x + 1

+5 +5

x = -2x + 6

+2x +2x

3x = 6

x = 2

y = x 5

y = (2) 5

y = 3

Answer: (2, 3)

8.

y = (1/2)x - 3

y = (3/2)x 1

y = (1/2)x 3

(3/2)x 1 = (1/2)x 3

+1 +1

(3/2)x = (1/2)x 4

(4/2)x = -4

2x = -4

x = -2

y = (3/2)x 1

y = (3/2)-2 1

y = -3 1

y = 4

Answer: (-2, 4)

9.

x + y = 2

4y = -4x + 8

x + y = 2

y = 2 x

4y = -4x + 8

4(2 x) = -4x + 8

8 4x = -4x + 8

+4x +4x

8 8x = 8

-8 -8

8x = 0

0 = 0

Answer: This equation has infinite solutions.

ELIMINATION METHOD

10.

y = (2/3)x - 1

y = -x + 4

0 = (-5/3)x + 5

(-5/3)x = 5

x = -3

y = -x + 4

y = 3 + 4

-3 -3

y = 1

Answer: (-3, 1)

11.

x + y = 0

3x + y = -4

2x + 0 = -4

2x = -4

x = -2

x + y = 0

-2 + y = 0

+2 +2

y = 2

Answer: (-2, 2)

12.

4x + 3y = -15

y = x + 2

y = x + 2 (times equation by 3)

3y = 3x + 6

4x + 3y = -15

7x = -21

x = -3

y = x + 2

y = -3 + 2

y = -1

Answer: (-3, -1)

13.

x + 2y = -4

4y = 3x + 12

x + 2y = -4 (times equation by 3)

3x + 6y = -12

4y = 3x + 12

3x - 4y = 3x 3x + 12

4y 3x = 12

3x + 6y = -12

3x + 6y + 4y 3x = -12 + 12

6y + 4y = -12 + 12

10y = 0

y = 0

x + 2y = -4

x + 2(0) = -4

x = -4

Answer: (-4, 0)

14.

y = 2x

x + y = 3

x + y = 3 (times equation by 2)

2x + 2y = 6

y = 2x

y 2x = 2x 2x

y 2x = 0

y 2x = 0

2x + 2y = 6

2x + 2y + y 2x = 6 + 0

2y + y = 6 + 0

3y = 6

y = 2

y = 2x

(2) = 2x

2x = 2

x = 1

Answer: (1, 2)

15.

x = 3 - 3y

x + 3y = -6

x + 3y = 3 3y + 3y

x + 3y = 3

x + 3y = 3

x + 3y = -6

Answer: Both equations are the same, but both equal two different things. Therefore the answer is no solution.

16.

y = -2x + 1

y = x 5

0 = -x + -4

x = -4

y = -2x + 1

y = -2(-4) + 1

y = 8 + 1

y = 9

Answer: (-4, 9)

17.

y = (1/2)x - 3

y = (3/2)x 1

0 = -x 4

x = 2

y = x 5

y = (-2) 5

y = 7

Answer: (2, -3)

18.

x + y = 2

4y = -4x + 8

4y = -4x + 8

4y 4y = -4x 4y + 8

-4x 4y = 8

x + y = 2 (times equation by 4)

4x + 4y = 8

-4x + 4y = 8

4x + 4y = 8

8y = 16

y = 2

x + y = 2

x + (2) = 2

-2 -2

x = 0

Answer: (0, 2)

"The thing about quotes on the Internet is you cannot confirm their validity"

~Abraham Lincoln

Offline

**bob bundy****Administrator**- Registered: 2010-06-20
- Posts: 8,321

hi demha

Welcome to the forum.

LATER EDIT: As I checked these I began to think I'd seen them before but I was rushing to finish while you were still on-line.

Now I've had time to look carefully, I see the second 9 are the same as the first 9, just a different method. You should, of course, get the same answers either way, which may help you to sort out the corrections.

SUBSTITUTION METHOD

1.

y = (2/3)x - 1

y = -x + 4

y = (2/3) - 1

(-x + 4) = (2/3)x 1

(-1 1/2)x + 4 = -1

(-1 1/2)x = -5

x = 3

y = -x + 4

y = -3 + 4

y = 1

Answer: (3, 1)

CORRECT

2.

x + y = 0

3x + y = -4

x + y = 0

-x -x

y = 0 x

3x + (0 x) = -4

2x + 0 = -4

2x = -4

x = -2

x + y = 0

(-2) + y = 0

+2 +2

y = 0 + 2

y = 2

Answer: (-2, 2)

CORRECT

3.

4x + 3y = -15

y = x + 2

4x + 3(x + 2) = -15

4x + (3x + 6) = -15

7x + 6 = -15

-6 -6

7x = -21

x = -3

y = x + 2

y = (-3) + 2

y = -1

Answer: (-3, -1)

CORRECT

4.

x + 2y = -4

4y = 3x + 12

x + 2y = -4

x = -4 - 2y

4y = 3(-4 - 2y) + 12

4y = (-12 6y) + 12

10y = -12 + 12

10y = 0

y = 0

4y = 3x + 12

4(0) = 3x + 12

3x = 12

x = 4

Answer: (4, 0)

SOMETHING HAS GONE WRONG HERE. YOUR VALUES DON'T FIT THE EQUATIONS.

5.

y = 2x

x + y = 3

x + y = 3

x + (2x) = 3

3x = 3

x = 1

y = 2x

y = 2(1)

y = 2

Answer: (1, 2)

CORRECT

6.

x = 3 - 3y

x + 3y = -6

x + 3y = -6

(3 3y) + 3y = -6

3 + 0 = -6

3 = -6

Answer: There is no solution.

CORRECT

7.

y = -2x + 1

y = x 5

y = -2x + 1

(x 5) = -2x + 1

+5 +5

x = -2x + 6

+2x +2x

3x = 6

x = 2

y = x 5

y = (2) 5

y = 3

Answer: (2, 3)

CHECK Y AGAIN

8.

y = (1/2)x - 3

y = (3/2)x 1

y = (1/2)x 3

(3/2)x 1 = (1/2)x 3

+1 +1

(3/2)x = (1/2)x 4

(4/2)x = -4

2x = -4

x = -2

y = (3/2)x 1

y = (3/2)-2 1

y = -3 1

y = 4

Answer: (-2, 4)

CHECK Y AGAIN

9.

x + y = 2

4y = -4x + 8

x + y = 2

y = 2 x

4y = -4x + 8

4(2 x) = -4x + 8

8 4x = -4x + 8

+4x +4x

8 8x = 8

-8 -8

8x = 0

0 = 0

Answer: This equation has infinite solutions.

CORRECT

ELIMINATION METHOD

10.

y = (2/3)x - 1

y = -x + 4

0 = (-5/3)x + 5

(-5/3)x = 5

x = -3

y = -x + 4

y = 3 + 4

-3 -3

y = 1

Answer: (-3, 1)

THESE VALUES DON'T FIT THE EQUATIONS

11.

x + y = 0

3x + y = -4

2x + 0 = -4

2x = -4

x = -2

x + y = 0

-2 + y = 0

+2 +2

y = 2

Answer: (-2, 2)

CORRECT

12.

4x + 3y = -15

y = x + 2

y = x + 2 (times equation by 3)

3y = 3x + 6

4x + 3y = -15

7x = -21

x = -3

y = x + 2

y = -3 + 2

y = -1

Answer: (-3, -1)

CORRECT

13.

x + 2y = -4

4y = 3x + 12

x + 2y = -4 (times equation by 3)

3x + 6y = -12

4y = 3x + 12

3x - 4y = 3x 3x + 12

4y 3x = 12

3x + 6y = -12

3x + 6y + 4y 3x = -12 + 12

6y + 4y = -12 + 12

10y = 0

y = 0

x + 2y = -4

x + 2(0) = -4

x = -4

Answer: (-4, 0)

CORRECT

14.

y = 2x

x + y = 3

x + y = 3 (times equation by 2)

2x + 2y = 6

y = 2x

y 2x = 2x 2x

y 2x = 0

y 2x = 0

2x + 2y = 6

2x + 2y + y 2x = 6 + 0

2y + y = 6 + 0

3y = 6

y = 2

y = 2x

(2) = 2x

2x = 2

x = 1

Answer: (1, 2)

CORRECT

15.

x = 3 - 3y

x + 3y = -6

x + 3y = 3 3y + 3y

x + 3y = 3

x + 3y = 3

x + 3y = -6

Answer: Both equations are the same, but both equal two different things. Therefore the answer is no solution.

CORRECT

16.

y = -2x + 1

y = x 5

0 = -x + -4

x = -4

y = -2x + 1

y = -2(-4) + 1

y = 8 + 1

y = 9

Answer: (-4, 9)

CHECK AGAIN

17.

y = (1/2)x - 3

y = (3/2)x 1

0 = -x 4

x = 2

y = x 5

y = (-2) 5

y = 7

Answer: (2, -3)

CHECK AGAIN

18.

x + y = 2

4y = -4x + 8

4y = -4x + 8

4y 4y = -4x 4y + 8

-4x 4y = 8

x + y = 2 (times equation by 4)

4x + 4y = 8

-4x + 4y = 8

4x + 4y = 8

8y = 16

y = 2

x + y = 2

x + (2) = 2

-2 -2

x = 0

Answer: (0, 2)

CANNOT BE RIGHT. COMPARE WITH Q9

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Offline

**demha****Member**- Registered: 2012-11-25
- Posts: 195

Thanks for the welcome and super thanks for answering back! I knew that 1 - 9 & 10 - 18 were the same equations... but now I feel kind of stupid ._.

I should have known that I would get the same answers! =P

Here are the mistakes I redid:

4.

x + 2y = -4

4y = 3x + 12

x + 2y = -4

x = -4 - 2y

4y = 3(-4 - 2y) + 12

4y = (-12 6y) + 12

10y = -12 + 12

10y = 0

y = 0

x + 2y = -4

x + 2(0) = -4

x = -4

Answer: (4, 0)

---

7.

y = -2x + 1

y = x 5

y = -2x + 1

(x 5) = -2x + 1

+5 +5

x = -2x + 6

+2x +2x

3x = 6

x = 2

y = x 5

y = (2) 5

y = -3

Answer: (2, -3)

8.

y = (1/2)x - 3

y = (3/2)x 1

y = (1/2)x 3

(3/2)x 1 = (1/2)x 3

+1 +1

(3/2)x = (1/2)x 4

(4/2)x = -4

2x = -4

x = -2

y = (3/2)x 1

y = (3/2)-2 1

y = -3 1

y = -4

Answer: (-2, -4)

---

(Elimination Method)

10.

y = (2/3)x - 1

y = -x + 4

0 = (-5/3)x + 5

(- 5/3)x = 5

x = 3

y = 2/3x 1

y = 2/3(3) 1

y = 2 1

y = 1

Answer: (3, 1)

---

16.

y = -2x + 1

y = x 5

0 = 3x + 6

3x = 6

x = 2

y = x 5

y = 2 5

y = -3

Answer: (2, -3)

17.

y = (1/2)x - 3

y = (3/2)x 1

0 = -x 2

x = -2

y = (3/2)x 1

y = (3/2)-2 1

y = -3 1

y = -4

Answer: (-2, -4)

---

Now for number 18, I understand I am supposed to get the same answer as number 9 which is a "word solution." The answer should be infinate (0 = 0 for example). I have tried a few times but I just can't seem to get it. Would you mind helping me? Here is the equation yet again:

(Elimination Method)

18.

x + y = 2

4y = -4x + 8

"The thing about quotes on the Internet is you cannot confirm their validity"

~Abraham Lincoln

Offline

**bob bundy****Administrator**- Registered: 2010-06-20
- Posts: 8,321

hi demha

Here are the mistakes I redid:

4.

x + 2y = -4

4y = 3x + 12

x + 2y = -4

x = -4 - 2y

4y = 3(-4 - 2y) + 12

4y = (-12 6y) + 12

10y = -12 + 12

10y = 0

y = 0

x + 2y = -4

x + 2(0) = -4

x = -4

Answer: (4, 0)

YOU MEANT (-4,0) OF COURSE !

---

7.

y = -2x + 1

y = x 5

y = -2x + 1

(x 5) = -2x + 1

+5 +5

x = -2x + 6

+2x +2x

3x = 6

x = 2

y = x 5

y = (2) 5

y = -3

Answer: (2, -3)

CORRECT

8.

y = (1/2)x - 3

y = (3/2)x 1

y = (1/2)x 3

(3/2)x 1 = (1/2)x 3

+1 +1

(3/2)x = (1/2)x 4

(4/2)x = -4

2x = -4

x = -2

y = (3/2)x 1

y = (3/2)-2 1

y = -3 1

y = -4

Answer: (-2, -4)

CORRECT

---

(Elimination Method)

10.

y = (2/3)x - 1

y = -x + 4

0 = (-5/3)x + 5

(- 5/3)x = 5

x = 3

y = 2/3x 1

y = 2/3(3) 1

y = 2 1

y = 1

Answer: (3, 1)

CORRECT

---

16.

y = -2x + 1

y = x 5

0 = 3x + 6

3x = 6

x = 2

y = x 5

y = 2 5

y = -3

Answer: (2, -3)

CORRECT

17.

y = (1/2)x - 3

y = (3/2)x 1

0 = -x 2

x = -2

y = (3/2)x 1

y = (3/2)-2 1

y = -3 1

y = -4

Answer: (-2, -4)

CORRECT

---

Now for number 18, I understand I am supposed to get the same answer as number 9 which is a "word solution." The answer should be infinate (0 = 0 for example). I have tried a few times but I just can't seem to get it. Would you mind helping me? Here is the equation yet again:

(Elimination Method)

18.

x + y = 2

4y = -4x + 8

4y = -4x + 8

4y 4y = -4x 4y + 8

-4x 4y = 8

This is where you went wrong.

line 2 is ok

4y 4y = -4x 4y + 8

so

0 = -4x 4y + 8

Now what are you doing? If you add 4x and add 4y to each side you'll have

0 +4x + 4y = -4x+4x 4y + 4y + 8

4x + 4y = 8

Now the other equation was

x + y = 2

so multiply by 4

4x + 4y = 8

So now eliminating gives

0 = 0 as you wanted.

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Offline

**demha****Member**- Registered: 2012-11-25
- Posts: 195

Hello Bob!

So it was adding 4x & 4y to each side that would give me the correct answer. I see now. Then we have both similar equations then start the elimination to get 0 + 0 = 0 which in the end is 0 = 0: there is an infinate solution.

Thanks you very much Bob! I appreciate you taking your time helping me with this.

"The thing about quotes on the Internet is you cannot confirm their validity"

~Abraham Lincoln

Offline

**bob bundy****Administrator**- Registered: 2010-06-20
- Posts: 8,321

You are welcome.

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Offline

Pages: **1**