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Please Help!! I need help solving word problems

by Lori
(WASHINGTON)











































I have a lot troubles understanding word problems on where to start the solving process. Please help I need answers by tomorrow please show all work to help me understand the how.

1. In solving the equation (x + 1)(x – 2) = 54, Eric stated that the solution would be
x + 1 = 54 => x = 53 or (x – 2) = 54 => x = 56
However, at least one of these solutions fails to work when substituted back into the original equation. Why is that? Please help Eric to understand better; solve the problem yourself, and explain your reasoning.


2. If a stone is tossed from the top of a 190 meter building, the height of the stone as a function of time is given by h(t)=-9.8t^2-10t+190, where t is in seconds, and height is in meters. After how many seconds will the stone hit the ground? Round to the nearest hundredth’s place; include units in your answer.



3. Use the discriminant to determine whether the following equations have solutions that are: two different rational solutions; two different irrational solutions; exactly one rational solution; or two different imaginary solutions.
8x^2+2x+4=0

a)Two different irrational solutions
b)Two different imaginary solutions
c)Exactly one rational solution
d)Two different rational solutions


4. Solve using the substitution method. Show your work. If the system has no solution or an infinite number of solutions, state this.
18x + 6y = 78
12x + 54y = -48


5. Solve using the elimination method. Show your work. If the system has no solution or an infinite number of solutions, state this.
14x+18y =-54
-4x-14y = 42

Thank you,
Lori



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Sep 10, 2011
Quadratic & Simultaneous Equations
by: Staff

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Part VI

Solve the new equation for y

9y - (49/2)y = -27 + (147/2)

Multiply each side of the new equation by 2

9y - (49/2)y = -27 + (147/2)

2 * [9y - (49/2)y] = 2 * [-27 + (147/2)]

(2 * 9y) + [2 * (-49/2) * y] = [2 * (-27)] + [2 * (147/2)]

(2 * 9y) + [(2 / 2) * (-49) * y] = [2 * (-27)] + [(2 / 2) * (147)]

(2 * 9y) + [(1) * (-49) * y] = [2 * (-27)] + [(1) * (147)]

(18y) + (-49y) = (-54) + (147)

18y - 49y = -54 + 147

(18y - 49y) = (-54 + 147)

(-31y) = (93)

-31y = 93

Divide each side of the equation by -31

(-31y) / (-31) = (93) / (-31)

y * [(-31) / (-31)] = (93) / (-31)

y * [1] = (93) / (-31)

y = (93) / (-31)

y = -3


To find x, you can substitute -3 for y in either of the original equations.

-2x - 7y = 21 (this is the simplified version of equation 2)

-2x - [7 * (-3)] = 21

-2x - (-21) = 21

-2x + [(-1) * (-21)] = 21

-2x + [21] = 21

-2x + 21 = 21


Subtract 21 from each side of the equation:

-2x + 21 = 21

-2x + 21 - 21 = 21 - 21

-2x + 0 = 0

-2x = 0


Divide each side of the equation by -2

-2x = 0

(-2x) / (-2) = 0 / (-2)

x * [(-2) / (-2)] = 0 / (-2)

x * [1] = 0 / (-2)

x = 0 / (-2)

x = 0



the final answer to problem 5 is: x = 0, y = -3


check the solution by substituting the numerical values of x and y into both original equations

1st equation (original): 14x + 18y = -54

x = 0, y = -3

14x + 18y = -54

(14 * 0) + [(18) * (-3)] = -54

(0) + [-54] = -54

0 - 54 = -54

-54 = -54, OK


2nd equation (original): -4x - 14y = 42

x = 0, y = -3

-4x - 14y = 42

[(-4) * (0)] - [(14) * (-3)] = 42

[(0)] - [-42] = 42

[(0)] + [(-1) * (-42)] = 42

[(0)] + [42] = 42

0 + 42 = 42

42 = 42, OK


since both equations are in balance for the values x = 0 and y = -3, x and y are valid solutions


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Thanks for writing.

Staff
www.solving-math-problems.com

Sep 10, 2011
Quadratic & Simultaneous Equations
by: Staff

--------------------------------------------------------------------

Part V

You can view this solution graphically by opening the following link:

(1) If your browser is Firefox, click the link to VIEW the solution; or if your browser is Chrome, Internet Explorer, Opera, or Safari (2A) highlight and copy the link, then (2B) paste the link into your browser Address bar & press enter:

Use the Backspace key to return to this page:

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--------------------------------------------------

5. Solve using the elimination method. Show your work. If the system has no solution or an infinite number of solutions, state this.


14x + 18y = -54

-4x - 14y = 42


You can simplify the first equation by dividing each side of the equation by 2

14x + 18y = -54

(14x + 18y)/2 = (-54)/2

(14x)/2 + (18y)/2 = (-54)/2

(14/2)*x + (18/2)*y = (-54)/2

(7)*x + (9)*y = (-54)/2

7x + 9y = (-54)/2

7x + 9y = -27

You can simplify the second equation by dividing divide each side of the equation by 2

-4x-14y = 42

(-4x - 14y)/2 = (42)/2

(-4x)/2 - (14y)/2 = (42)/2

(-4/2)*x - (14/2)*y = (42)/2

(-2)*x - (7)*y = (42)/2

-2x - 7y = (42)/2


The simplified versions of both equations are:
7x + 9y = -27

-2x - 7y = 21


Since you will be using the substitution method to solve this system of equations, one approach is to eliminate the x variable first.

To accomplish this, change the terms of the second equation so the coefficient for the x variable is exactly the same as the coefficient for the x variable in equation 1, but with the opposite sign. You can accomplish this by multiplying the second equation by 7/2.

-2x - 7y = 21

(-2x - 7y) * (7/2) = (21) * (7/2)

(-2x) * (7/2) + (-7y) * (7/2) = (21) * (7/2)

(-2/1) * (7/2) * x + (-7/1) * (7/2) * y = (21) * (7/2)

(-2*7)/(1*2) * x + (-7*7)/(1*2) * y = (21) * (7/2)

(-2*7)/(1*2) * x + (-7*7)/(1*2) * y = (21) * (7/2)

(-14)/( 2) * x + (-49)/(2) * y = (21) * (7/2)

(-7) * x - (49/2) * y = (147/2)

-7x - (49/2)y = (147/2)


Both equations now look like this:

7x + 9y = -27

-7x - (49/2)y = (147/2)


You can now add these equations to eliminate the x variable

7x + 9y = -27

+ [-7x - (49/2)y = (147/2)]

----------------------------------

7x - 7x + 9y - (49/2)y = -27 + (147/2)

(7x - 7x) + 9y - (49/2)y = -27 + (147/2)

(0) + 9y - (49/2)y = -27 + (147/2)

9y - (49/2)y = -27 + (147/2)

The x variable does not appear in the new equation. It has been eliminated.


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Sep 10, 2011
Quadratic & Simultaneous Equations
by: Staff

--------------------------------------------------------------------

Part IV

the discriminant is

Δ = (b² - 4ac)


Δ = (2² - 4*8*4)

Δ = (4 - 128)

Δ = -124

Choice b): There are two imaginary solutions since √(b² - 4ac) = √(-124) ≈ ±11.1355 i



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4. Solve using the substitution method. Show your work. If the system has no solution or an infinite number of solutions, state this.

18x + 6y = 78

12x + 54y = -48


You can simplify the second equation by dividing each side of the equation by 6

(12x + 54y)/6 = (-48)/6

12x/6 + 54y/6 = (-48)/6

(12/6)*x + (54/6)*y = (-48)/6

(2)*x + (9)*y = -8

2x + 9y = -8



You can simplify the first equation by dividing each side of the equation by 6

(18x + 6y)/6 = (78)/6

18x/6 + 6y/6 = 78/6

(18/6)*x + (6/6)*y = 78/6

(3)*x + (1)*y = 13

3x + y = 13


Solve the simplified version of the 1st equation for y

3x + y = 13

3x - 3x + y = 13 - 3x

0 + y = 13 - 3x

y = 13 - 3x


SUBSTITUTE the SOLUTION of y in the 2nd equation

y = 13 - 3x (is the solution for y from the 1st equation)

2x + 9y = -8 (this is the simplified version of the 2nd equation)


Substitute 13 - 3x for y in simplified version of the 2nd equation

2x + 9y = -8

2x + 9(13 - 3x) = -8


Solve for x

Eliminate the parentheses on the left side of the equation using the distributive law

2x + 9(13 - 3x) = -8

2x + (9)*(13) + (9)*(-3x) = -8

Combine like terms

2x + 117 - 27x = -8

2x - 27x + 117 = -8

(2x - 27x) + 117 = -8

(-25x) + 117 = -8

Add 25x to each side of the equation

-25x + 117 = -8

-25x + 25x + 117 = -8 + 25x

(-25x + 25x) + 117 = -8 + 25x

0 + 117 = -8 + 25x

117 = -8 + 25x

Add 8 to each side of the equation

117 + 8 = -8 + 25x + 8

125 = -8 + 25x + 8

125 = -8 + 8 + 25x

125 = (-8 + 8) + 25x

125 = 0 + 25x

125 = 25x

Divide each side of the equation by 25

125 / 25 = 25x / 25

5 = 25x / 25

5 = x * (25 / 25)

5 = x * (1)

5 = x

x = 5


Now that you know that x = 5, substitute the number 5 wherever x appears in either of the equations.

3x + y = 13 (this is the simplified version of the 1st equation)

x = 5

3 * 5 + y = 13

15 + y = 13


Solve for y

15 + y = 13

15 + y - 15 = 13 - 15

15 - 15 + y = 13 - 15


(15 - 15) + y = 13 - 15

0 + y = 13 - 15

y = 13 - 15

y = -2


the final answer to problem 4 is: x = 5, y = -2


check the solution by substituting the numerical values of x and y into both original equations

1st equation (original): 18x + 6y = 78

x = 5, y = -2

18x + 6y = 78

18 * 5 + 6 * (-2) = 78

90 - 12 = 78

78 = 78, OK

2nd equation (original): 12x + 54y = -48

x = 5, y = -2

12x + 54y = -48

12 * 5 + 54 * (-2) = -48

60 + 54 * (-2) = -48

60 - 108 = -48

-48 = -48, OK

since both equations are in balance for the values x = 5 and y = -2, x and y are valid solutions


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Sep 10, 2011
Quadratic & Simultaneous Equations
by: Staff

--------------------------------------------------------------------

Part III


Solve for t

The equation [0 = -9.8t² - 10t + 190] is a quadratic equation

0 = -9.8t² - 10t + 190

-9.8t² - 10t + 190 = 0

Multiply each side of the equation by -1

(-1)(-9.8t² - 10t + 190) = (-1)(0)

(-1) * (-9.8t²) + (-1) * (-10t) + (-1) * (190) = (-1) * (0)

9.8t² + 10t - 190 = 0

At this point you can choose what approach you will employ to solve the quadratic equation:

(1) Solve by Factoring (factoring is not a feasible approach for this problem)
(2) Use the Quadratic Formula
(3) Complete the Square
(4) Use the Indian Method for solving a quadratic equation (similar to completing the square)
(5) Solve the equation graphically

You will probably find it easiest to use the Quadratic Formula to solve this problem.

Quadratic Formula

ax² + bx + c = 0

x = [-b ± √(b² - 4ac)]/(2a)


9.8t² + 10t - 190 = 0

t = [-10 ± √(10² - 4*9.8*(-190)]/(2*9.8)

t = [-10 ± √(7548)]/(19.6)

t ≈ [-10 ± 86.8792]/(19.6)


1st solution

t ≈ [-10 + 86.8792]/(19.6)

t ≈ 3.92241


2nd solution

t ≈ [-10 - 86.8792]/(19.6)

t ≈ -4.94282

However, the 2nd solution is invalid. Even though t ≈ -4.94282 seconds is a valid solution to the quadratic equation 9.8t² + 10t - 190 = 0, it is NOT a VALID SOLUTION TO THE PROBLEM.

The time it takes for the stone to fall to the ground cannot be a negative time.

The 2nd solution cannot be used.


the final answer to problem 2 is: t ≈ 3.92 seconds



check the solution by substituting the numerical value for t into the original equation:

solution, t ≈ 3.92241 seconds

-9.8t² - 10t + 190 ≈ 0

(-9.8)(3.92241²) - (10)(3.92241) + 190 ≈ 0

(-9.8)(15.3853) - (10)(3.92241) + 190 ≈ 0

-150.776 - 39.2241 + 190 ≈ 0

-190 + 190 ≈ 0

0 = 0, OK → t ≈ 3.92241 seconds is a valid solution


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3. Use the discriminant to determine whether the following equations have solutions that are: two different rational solutions; two different irrational solutions; exactly one rational solution; or two different imaginary solutions.

a)Two different irrational solutions
b)Two different imaginary solutions
c)Exactly one rational solution
d)Two different rational solutions

8x² + 2x + 4 = 0


x = [-b ± √(b² - 4ac)]/(2a)

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Sep 10, 2011
Quadratic & Simultaneous Equations
by: Staff

--------------------------------------------------------------------

Part II


I am going to solve this equation by factoring because I think it will be easier for you to compare with the original question.

Eric (see the statement of the problem) tried to use factoring, but failed because the equation he used [(x + 1)(x - 2) = 54] was not in the general quadratic form.

You can factor the left hand side of the equation like this:

x² - x - 56 = 0

(x + 7)(x - 8) = 0


There are two solutions:

(x + 7)(x - 8) = 0


1st solution:

"Divide" each side of the equation "by (x - 8)", and then solve for x

(x + 7)(x - 8) = 0

(x + 7)(x - 8) / (x - 8) = 0 / (x - 8)

(x + 7) * [(x - 8) / (x - 8)] = 0 / (x - 8)

(x + 7) * [1] = 0/(x - 8)

(x + 7) = 0/(x - 8)

x + 7 = 0/(x - 8)

x + 7 = 0

x + 7 - 7 = 0 - 7

x + 0 = 0 - 7

x = 0 - 7

x = -7



2nd solution:

"Divide" each side of the equation "by (x + 7)", and then solve for x

(x + 7)(x - 8) = 0

(x - 8)(x + 7) / (x + 7) = 0 / (x + 7)

(x - 8) * [(x + 7) / (x + 7)] = 0 / (x + 7)

(x - 8) * [1] = 0/( x + 7)

(x - 8) = 0/(x + 7)

x - 8 = 0/(x + 7)

x - 8 = 0

x - 8 + 8 = 0 + 8

x + 0 = 0 + 8

x = 0 + 8

x = 8


the final answer to problem 1 is: x ∈{-7, 8}



check the solution by substituting the two numerical values of x into the original equation:

1st solution, x = -7

(x + 1)(x - 2) = 54

(-7 + 1)(-7 - 2) = 54

(-6)(-9) = 54

54 = 54, OK → x = -7 is a valid solution



2nd solution, x = 8


(x + 1)(x - 2) = 54

(8 + 1)(8 - 2) = 54

(9)(6) = 54

54 = 54, OK → x = 8 is a valid solution


You can view this solution graphically by opening the following link:

(1) If your browser is Firefox, click the link to VIEW the solution; or if your browser is Chrome, Internet Explorer, Opera, or Safari (2A) highlight and copy the link, then (2B) paste the link into your browser Address bar & press enter:

Use the Backspace key to return to this page:

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2. If a stone is tossed from the top of a 190 meter building, the height of the stone as a function of time is given by h(t)=-9.8t^2-10t+190, where t is in seconds, and height is in meters. After how many seconds will the stone hit the ground? Round to the nearest hundredth's place; include units in your answer.

h(t) = -9.8t² - 10t + 190


The are two variables:

h(t) = height above the ground = 0 (the stone's height above the ground when the stone hits the ground is 0 because the stone hits the ground at ground level)

t = time in seconds = unknown


Substitute any known values in the equation (the only known value is h(t) = 0)

h(t) = -9.8t² - 10t + 190

0 = -9.8t² - 10t + 190

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Sep 10, 2011
Quadratic & Simultaneous Equations
by: Staff

Part I

The question:

by Lori
(WASHINGTON)

I have a lot troubles understanding word problems on where to start the solving process. Please help I need answers by tomorrow please show all work to help me understand the how.

1. In solving the equation (x + 1)(x - 2) = 54, Eric stated that the solution would be
x + 1 = 54 => x = 53 or (x - 2) = 54 => x = 56
However, at least one of these solutions fails to work when substituted back into the original equation. Why is that? Please help Eric to understand better; solve the problem yourself, and explain your reasoning.


2. If a stone is tossed from the top of a 190 meter building, the height of the stone as a function of time is given by h(t)=-9.8t^2-10t+190, where t is in seconds, and height is in meters. After how many seconds will the stone hit the ground? Round to the nearest hundredth's place; include units in your answer.



3. Use the discriminant to determine whether the following equations have solutions that are: two different rational solutions; two different irrational solutions; exactly one rational solution; or two different imaginary solutions.
8x^2+2x+4=0

a)Two different irrational solutions
b)Two different imaginary solutions
c)Exactly one rational solution
d)Two different rational solutions


4. Solve using the substitution method. Show your work. If the system has no solution or an infinite number of solutions, state this.
18x + 6y = 78
12x + 54y = -48


5. Solve using the elimination method. Show your work. If the system has no solution or an infinite number of solutions, state this.
14x+18y =-54
-4x-14y = 42

Thank you,
Lori


The answer:

--------------------------------------------------

1. (x + 1)(x - 2) = 54

(x + 1)(x - 2) = 54

First, you should rewrite the equation in the "general quadratic form" (not the standard form, the general form). It should look like this:

ax² + bx + c = 0


To rewrite in its general form, begin by expanding this expression on the left side of the equation [multiply (x + 1) by (x - 2)]

(x + 1)(x - 2) = 54

x² - x - 2 = 54

To remove 54 from the right side of the equation, subtract 54 from each side of the equation:

x² - x - 2 - 54 = 54 - 54

x² - x - 56 = 54 - 54

x² - x - 56 = 0

The equation is now in the general quadratic form:

x² - x - 56 = 0

At this point, you have at least 5 choices on how to proceed to the solution:

(1) Solve by Factoring
(2) Use the Quadratic Formula
(3) Complete the Square
(4) Use the Indian Method for solving a quadratic equation (similar to completing the square)
(5) Solve the equation graphically

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