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x = one number
x - 12 = other number {one number is 12 less than the other}

x + x - 12 = 72 {sum of the two numbers is 72}
2x - 12 = 72 {combined like terms}
2x = 84 {added 12 to each side}
x = 42 {divided each side by 2}
x - 12 = 30 {substituted 30, in for x, into x - 12}

42 and 30 are the two numbers
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There are 52 coins in a purse. Some of them are quarters, and some are dimes.
The total value of all coins is $7.45. how many were quarters and how many were dimes?

x = number of quarters
52 - x = number of dimes {there are 52 coins total}

0.25x + 0.1(52 - x) = 7.45 {number of coins times value of coin equals total value}
0.25x + 5.2 - 0.1x = 7.45 {used distributive property}
0.15x + 5.2 = 7.45 {combined like terms}
0.15x = 2.25 {subtracted 5.2 from each side}
x = 15 {divided each side by 0.15}
52 - x = 37 {substituted 15, in for x, into 52 - x}

15 quarters and 37 dimes in the purse
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8x - 5 = 35
8x = 40 {added 5 to each side}
x = 5 {divided each side by 8}
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When it hits the ground, the height will be zero.

h(t) = 127.4 - 4.9t²
0 = 127.4 - 4.9t² {substituted 0 in for the height, h(t)}
4.9t² = 127.4 {added 4.9t² to each side}
t² = 26 {divided each side by 4.9}
t ≈ 5.1 {took the square root of each side and rounded}
In approximately 5.1 seconds, it will hit the ground.
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x^4yz^3
-------
xyz^6

x³
-- {when dividing same variables with exponents, think of subtract the exponents or cancel}
z³
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x + x + 6 = 14
2x + 6 = 14 {combined like terms}
2x = 8 {subtracted 6 form each side}
x = 4 {divided each side by 2}
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x = height
2x + 1 = length {length is 1 more than twice its height}

Area of a rectangle is length x height

x(2x + 1) = 351 {multiplied length x height and set equal to area, 351}
2x² + x = 351
2x² + x - 351 = 0
2x² - 26x + 27x - 351 = 0 {split x into -26x and 27x}
2x(x - 13) + 27(x - 13) = 0 {factored 2x out of 1st two terms and 27 out of last two terms - factoring by grouping}
(2x + 27)(x - 13) = 0 {factored (x - 13) out of the two terms}
2x + 27 = 0 or x - 13 = 0 {set each factor equal to 0}
2x = -27 or x = 13 {subtracted 27 in 1st and added 13 in 2nd}
x = -13.5 or x = 13 {solved each equation for x}
x = 13 {height of a rectangle cannot be negative}
2x + 1 = 27 {substituted 13, in for x, into 2x + 1}

height is 13 cm and length is 27 cm

Perimeter of a rectangle is 2(height) + 2(length)
P = 2(13) + 2(27) {substituted 13 for height and 27 for length, into perimeter formula}
= 26 + 54 {multiplied}
= 80 {added}

Perimeter of rectangle is 81 cm
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(-0.4)³
= (-0.4)(-0.4)(-0.4) {took (-0.4) to the third power}
= -0.064 {multiplied}
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Find three consecutive odd integers, so that 3 times their sum is 5 more than 8 times the middle one.

x = 1st odd integer
x + 2 = 2nd odd integer
x + 4 = 3rd odd integer {odd and also even integers increase by 2}

3(x + x + 2 + x + 4) = 8(x + 2) + 5 {sum is equal to 5 more than 8 times the middle one}
3(3x + 6) = 8x + 16 + 5 {combined like terms on left, used distributive property on right}
9x + 18 = 8x + 21 {combined like terms}
x = 3 {subtracted 8x and subtracted 18 from each side}
x + 2 = 5 {substituted 3, in for x, into x + 2}
x + 4 = 7 {substituted 3, in for x, into x + 4}

3,5, and 7 are the three consecutive odd integers
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Domain is the set of x-values {the input}

The graph of the function f(x) = x² + 2x + 1 is a parabola {u-shaped graph}
opening upward

The domain is the set of all real numbers.
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A right triangle is formed with two legs of 6√2 ft and the diagonal as the hypotenuse.

a² + b² = c² {the Pythagorean Theorem}
(6√2)² + (6√2)² = c² {substituted 6√2 in for the legs, a and b}
36(2) + 36(2) = c² {evaluated the exponents}
72 + 72 = c² {multiplied}
144 = c² {added}
c = 12 {took the square root of each side}
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x = Mary's age
x + 5 = George's age

x + x + 5 = 43 {sum of ages is 43}
2x + 5 = 43 {combined like terms}
2x = 38 {subtracted 5 from each side}
x = 19 {divided each side by 2}
x + 5 = 24 {substituted 19, in for x, into x + 5}

Mary is 19
George is 24
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-√-98
= -√-1 √49 √2 {factored -98 into -1, 49, and 2}
= -7i√2 {when working with complex numbers, √-1 is represented by i}
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x = width
4x = length {it is four times as long as it is wide}

Perimeter of a rectangle = 2(width) + 2(length)

2(x) + 2(4x) = 100 {substituted into perimeter formula}
2x + 8x = 100 {multiplied}
10x = 100 {combined like terms}
x = 10 {divided each side by 10}
4x = 40 {substituted 10, in for x, into 4x}

length = 40 m
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x = starting number of boxes

x + 7
------ = 22 {half of all boxes is 22}
2

x + 7 = 44 {multiplied each side by 2}
x = 37 {subtracted 7 from each side}


She started with 37 boxes
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x = starting number of boxes

x + 7
------ = 22 {half of all boxes is 22}
2

x + 7 = 44 {multiplied each side by 2}
x = 37 {subtracted 7 from each side}

She started with 37 boxes
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√(49x)³
= √49x² √x {grouped the perfect squares under the first square root sign}
= 7x √x {evaluated the square root of 49x²}
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(7B)^-2

1
----- {to make a negative exponent positive, move the term up or down, opposite of where it currently is}
(7B)²

1
----- {took everything in parentheses to the 2nd power}
49B²
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4(2x + 3) + 2(x + 5) = 7
8x + 12 + 2x + 10 = 7 {used distributive property}
10x + 22 = 7 {combined like terms}
10x = -15 {subtracted 22 from each side}
x = -15/10 {divided each side by 10}
x = -3/2 {reduced}
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3c + 2d = 2
d = 4

3c + 2(4) = 2 {substituted 4 for d}
3c + 8 = 2 {multiplied}
3c = -6 {subtracted 8 from each side}
c = -2 {divided each side by 3}

c = -2 and d = 4
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x = the number

(x + 1)(x - 1) = 24 {one more than a number times one less than a number is 24}
x² - 1 = 25 {used foil method}
x² = 25 {added 1 to each side}
x = 5 or -5 {took the square root of each side}

the number could be 5 or -5
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Pythagorean Theorem
a² + b² = c² {c is the hypotenuse, a and b are the legs}
8² + b² = 17² {substituted given values into Pythagorean Theorem}
64 + b² = 289 {evaluated exponents}
b² = 225 {subtracted 64 from each side}
b = 15 {took the square root of each side}
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(2 x 10^4) x (3 x 10^6)
= (2 x 3) x (10^4 x 10^6) {grouped terms together}
= 6 x 10^10 {multiplied and added exponents on same base, when multiplying}
= 60,000,000,000 {moved decimal point to the right 10 places}
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x = one weight
2x - 7 = other weight {one is 7 less than twice the other}

x + 2x - 7 = 38 {the combined weight of the two is 38}
3x - 7 = 38 {combined like terms}
3x = 45 {added 7 to each side}
x = 15 {divided each side by 3}
2x - 7 = 23 {substituted 23, in for x, into 2x - 7}

15 kg and 23 kg are the weights of each piece
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Difference between 36 and 24 is 12
12 is twice 6
6 is the square root of 36

36 is the number
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N is Jerry's age

2N = Tom's age {Tom is equal to two times Jerry's age}
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4a² - 12a - 16 = 0 {I assume you meant equal to 0}
4(a² - 3a - 4) = 0 {factored 4 out}
4(a - 4)(a + 1) = 0 {factored into two binomials}
a - 4 = 0 or a + 1 = 0 {set each factor equal to 0, excluding the 4}
a = 4 or a = -1 {solved each equation for a}
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2x + 2y = 62 {perimeter is 2 times width + 2 times length}
xy = 238 {area is width times length}

x + y = 31 {divided each side of top equation by 2}
x = -y + 31 {subtracted y from each side}

y(-y + 31) = 238 {substituted -y + 31, in for x, into x(y) = 238}
-y² + 31y = 238 {used distributive property}
y² - 31y + 238 = 0 {added y² and subtracted 31y from each side}
(y - 17)(y - 14) = 0 {factored into two binomials}
y - 17 = 0 or y - 14 = 0 {set each factor equal to 0}
y = 17 or y = 14 {solved each equation for x}

width is 14 cm and length is 17 cm
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x - 7
------ = 1/4
x + 2

4(x - 7) = 1(x + 2) {cross-multiplied}
4x - 28 = x + 2 {used distributive property}
3x = 30 {subtracted x and added 28 to each side}
x = 10 {divided each side by 3}
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Parallel lines have equal slopes

Slope-intercept form is y = mx + b
m is the slope
b is the y-intercept

x - y = 4
-y = -x + 4 {subtracted x from each side}
y = x - 4 {divided each side by -1}
slope is 1

point is (-3,2) slope is 1
y - y1 = m(x - x1) {point-slope form}
y - 2 = 1[x - (-3)] {substituted into point-slope form}
y - 2 = x + 3 {simplified}
-x + y = 5 {put into standard form}
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216 km is 216,000 meters {1 km is 1,000 m}
3 hrs is 10,800 s {1 hour is 3,600 seconds}

216,000
-------- {divided meters by seconds}
10,800

= 20 m/s {divided}
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