To find (f o h)(4), substitute h(4) into f(x) and evaluate the expression.
Explanation:To find (f o h)(4), we need to first find h(4) and then substitute that value into f(x). Let's start by finding h(4).
h(x) = - x + 3
h(4) = -4 + 3 = -1
Now, we substitute h(4) into f(x).
f(x) = -3x - 1
f(h(4)) = f(-1) = -3(-1) - 1 = 2
Therefore, (f o h)(4) = 2.
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Angle addition and angle bisectors. Can someone help me with this please?
Answer:
ABE is 75
Step-by-step explanation:
If ABD = 108, that is the entire picture.
CBD is the part that is bisected. Bisector line BE cuts the angle in half.
Since CBD is 66, half of it is what is pictured on both sides of line BE.
Half of 66 is 33.
Angles EBE and EBD are both 33.
Angle ABE is ABC + CBE.
Angle ABE is also ABD - EBD.
ABE = ABD - EBD
ABE = 108 - 33
ABE = 75
7. 6h - 18 < 6h + 1
Answer:
Explanation:In this exercise we have to solve the following inequality:
[tex]6h-18<6h+1[/tex]
Step 1. Subtract 6h from both sides:
[tex]6h-18-6h<6h+1-6h[/tex]
Step 2. Simplify:
[tex]6h-6h-18<6h-6h+1 \\ \\ -18<1[/tex]
This is always true because -18 is a number less than 1. Therefore, for any real h-value the inequality is always true.
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Tom simplified an expression in three steps, as shown:
Which is the first incorrect step and why?
Step 1, all the exponents are increased by 4
Step 2, the exponents of the same base are added during multiplication
Step 3, the exponents of the same base are subtracted during division
Step 3, all the exponents are added during division
Answer:
Step 1) is incorrect.
Step-by-step explanation:
Answer:
Step one is the one that is wrong
Step-by-step explanation:
Cat hacker
Which statements are true given the histogram? Check
all that apply.
The histogram is symmetrical.
The histogram is evenly distributed.
The cluster from 4 p.m.-10 p.m. means that most of
the texts were sent in the afternoon and evening.
U More texts were sent from 6 p.m.-8 p.m. than from 8
p.m.-10 p.m.
The peak from 4 p.m.-6 p.m. means that the highest
number of texts was sent during this interval.
10 pm-1159
Answer:
C and E
Step-by-step explanation:
The cluster from 4 p.m.–10 p.m. means that most of the texts were sent in the afternoon and evening.
The peak from 4 p.m.–6 p.m. means that the highest number of texts was sent during this interval.
How many solutions does this system have?
Answer:
Infinitely many solutions.
Step-by-step explanation:
3x-5y=15
6x-10y=30
----------------
simplify 6x-10y=30 into 3x-5y=15
Answer:
C, Since there is is more then one solution
Step-by-step explanation:
The local appliance store is offering 15% off an entire purchase of $2,000 or more; or 30% off a single
item of $1,000 or more. Use the prices listed above to determine which offer will save the most
money on these appliances?
Answer:
The offer of 30% off a single item of $ 1000 or more will save most money on these appliances .
Step-by-step explanation:
Given as :
The purchase price = $ 2000
The offering discount = 15 %
Let The price of item after discount = x
So , x = ( $ 2000 ) - ( $ 2000 × 15 % )
Or, x = ( $ 2000 ) - ( $ 2000 × 0.15 )
Or, x = ( $ 2000 ) - ( $ 300 )
∴ x = $ 1700
Now, Again
The purchase price = $ 1000
The offering discount = 30 %
Let The price of item after discount = y
So , y = ( $ 1000 ) - ( $ 1000 × 30 % )
Or, y = ( $ 1000 ) - ( $ 1000 × 0.3 )
Or, y = ( $ 1000 ) - ( $ 300 )
∴ y = $ 700
Now, If we purchase two item at the cost price of $ 700 each , then it will cost as $ 700 × 2 = $ 1400
And for the same number of items at the cost price is $ 1700 after 15 % discount on list price of $ 2000 , So the second offer will save most .
Hence The offer of 30% off a single item of $ 1000 or more will save most money on these appliances . Answer
What is the value of x in the equation? 5 5/6 x + 2 2/3 = 12
Answer:
The value of x is 8/5
Step-by-step explanation:
(35/6)x + (8/3) = 12
(35/6)x = 28/3 (cross multiplication)
105x = 168
x = 8/5
Answer: 1 3/5 or 8/5
Step-by-step explanation:
(X+5) with exponent 1/2 equals (5-2x) with exponent 1/4
Answer: x=-2
Step-by-step explanation: Eliminate the fractional exponent by raising both sides of the equation by the reciprocal of the exponent.
Hope this helps you out! ☺
Answer:
x = -2.
Step-by-step explanation:
(x + 5)^1/2 = (5 - 2x)^1/4
Taking both sides to the power of 4:
(x + 5)^2 = 5 - 2x
x^2 + 10x + 25 = 5 - 2x
x^2 + 12x + 20 = 0
(x + 10(x + 2) = 0
x = -2, -10
Test for extraneous roots.
When x = -10
(-10 + 5)^1/2 = (-5)^1/2 which is not real.
When x = -2
(-2+5)^1/2 = 3 ^1/2
(5 - 2*(-2)) ^1/4 = 9 ^ 1/4 = 3^1/2
so x = -2 is a root.
John draws 15 cards from a deck of cards, if the card drawn is Queen, King, or Jack it worth 10 points
others worth 5 points. If he receives a total of 125 point. How many cards of each he draws from the
deck.
Answer:
10 face cards and 5 non-face cards
Step-by-step explanation:
If x is the number of face cards, and y is the number of non-face cards, then:
x + y = 15
10x + 5y = 125
Solve the system of equations. Using substitution:
10x + 5 (15 − x) = 125
10x + 75 − 5x = 125
5x = 50
x = 10
y = 15 − x
y = 5
He drew 10 face cards and 5 non-face cards.
Solve by factoring 2x^2+13x-7=0
Answer:
x=-7 and (1/2)
Step-by-step explanation:
anybody help me to solve this question.
Answer:
[tex]\dfrac{1}{(b-c)},\dfrac{1}{(c-a)} ,\dfrac{1}{(a-b)} are\ in\ AP[/tex]
Step-by-step explanation:
Given that [tex](b-c)^2, (c-a)^2 , (a-b)^2[/tex] are in AP
To prove: [tex]\dfrac{1}{(b-c)},\dfrac{1}{(c-a)} ,\dfrac{1}{(a-b)}[/tex] are in AP
From given as we know if p , q, r are in AP then 2q= p+r.
[tex]2(c-a)^2= (b-c)^2+(a-b)^2\\\\\Rightarrow 2(c^2+a^2-2ac)=b^2+c^2-2bc+a^2+b^2-2ab\\\\\Rightarrow 2c^2+2a^2-4ac= 2b^2+c^2+a^2 -2bc-2ab\\\\\Rightarrow a^2+c^2-2b^2-4ac= -2bc-2ab\\\\\Rightarrow a^2-2b^2+c^2= 4ac-2bc-2ab[/tex]
Now
[tex]\dfrac{1}{(b-c)},\dfrac{1}{(c-a)} ,\dfrac{1}{(a-b)}2\dfrac{1}{(c-a)} =\dfrac{1}{(b-c)}+\dfrac{1}{(a-b)}\\\\\Rightarrow \dfrac{2}{(c-a)}= \dfrac{a-b+b-c}{(b-c)(a-b)} \\\\\Rightarrow \dfrac{2}{(c-a)}= \dfrac{a-c}{(b-c)(a-b)} \\\\\Rightarrow2(b-c)(a-b) = (c-a)(a-c) \\\\\Rightarrow 2(ab-b^2-ac+bc)= -(a-c)^2\\\\\Rightarrow 2ab- 2b^2-2ac+2bc = -a^2-c^2+2ac\\\\\Rightarrow a^2-2b^2+c^2=4ac-2ab-2bc[/tex]
Which is the result of AP
.
Hence proved
In a finsh tank, 75% is female finshes. The rest are male. Female fishes are also 5 more than twice
of the male fishes. How many female and male fishes in the tank?
(Set up the rational equation, and solve for x!) PLEASE HELP ME
Answer:
5 Male fishes, and 15 Female fishes
Step-by-step explanation:
We have two unknowns: The number of lady fishes (let's represent this number with "L"), and the number of male fishes (let's name this unknown number "M"). So now we need to create two equations to help us solve for these unknowns.
Use the actual sentences they give you to create the equations:
a) "75% of the total number of fishes are female"
Notice that the total number of fishes is the addition of the number of ladies plus the number of males; that is the total is "L+M". According to the sentence, 75% (which in math terms is written as "0.75") of this total number equals the number of lady fishes:
[tex]0.75 \,*\,(L+M)=L[/tex]
now we work a little the algebra indicated in this equation to simplify it a bit:
[tex]0.75 \,*\,(L+M)=L\\0.75\,L+0.75\,M=L\\0.75\,M=L-0.75\,L\\0.75\,M=0.25\,L\\L=\frac{ 0.75\,M}{0.25}\\L=3\,M[/tex]
which tells us that the number of lady fishes is 3 times thenumber of male fishes.
b) Now the second equation based on the words:
"Lady fishes are 5 more than twice the male fishes"
"Twice the male fishes" can be written as: "2M"
so now we write this sentence as:
[tex]L=2\,M+5[/tex]
Now we combine the two equations by using in the second one the replacement for "L" in terms of "M" that we found in the first equation (that is: "L = 3 M") , so we obtain an equation with just one unknown (M) and we can solve for it:
[tex]L=2\,M+5\\3\,M=2\,M+5\\3\,M-2\,M=5\\M=5[/tex]
Which tells us that the number of male fishes in the tank is "5".
We can use now our first simplified equation to find the number of lady fishes:
[tex]L=3\,M\\L=3\,(5)\\L=15[/tex]
So, there are 15 female fishes in the tank.
If a figure is translated 7units to the right and 4 units down what can be said about the perimeter of the transformed image
Answer:
nothing has changed about the shape
Step-by-step explanation:
the shapes perimeter has neither grown or shrunk as it wasn't dialated only moved
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The volume of gas in a container is 125,000
liters, and the pressure is 1.2 atmospheres. Suppose the temperature remains constant, and the pressure changes to 1.6 atmospheres.
What is the new volume of the gas in liters?
Answer:
93750 L
Step-by-step explanation:
P1V1 = P2V2
1.2 atm*125000L = 1.6 atm* V2
V2 = 93750 L
The new volume of the gas is 93750 liters.
Given,
The volume of gas in a container is 125,000 liters.
The pressure is 1.2 atmospheres.
Suppose the temperature remains constant.
The pressure changes to 1.6 atmospheres.
We have to find the new volume of the gas.
What is the ideal gas law equation?The equation is given by:
PV = nRT
Where P is the pressure of the gas.
V is its volume.
n is the number of moles of the gas.
T is its kelvin temperature.
R is the ideal (universal) gas constant.
Let the initial volume and pressure of the gas be [tex]V_{1}[/tex] and [tex]P_{1}[/tex].
The final volume and pressure of the gas be [tex]V_{2}[/tex] and [tex]P_{2}[/tex].
We have,
[tex]V_{1}[/tex] = 125000 litres.
[tex]P_{1}[/tex] = 1.2 atmospheres.
Since temperature remains constant.
[tex]P_{1} V_{1}[/tex] / nR = T and [tex]P_{2} V_{2}[/tex] / nR = T
We can write as:
[tex]P_{1} V_{1}[/tex] / nR = [tex]P_{2} V_{2}[/tex] / nR
[tex]P_{1} V_{1}[/tex] = [tex]P_{2} V_{2}[/tex]
Putting the values we get,
[tex]P_{1} V_{1}[/tex] = [tex]P_{2} V_{2}[/tex]
1.2 x 125000 = 1.6 x [tex]V_{2}[/tex]
[tex]V_{2}[/tex] = (1.2 x 125000) / 1.6
= 93750 litres.
Thus the new volume of the gas is 93750 liters.
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Which function represents the sequence: 4, 7, 10, 13, 16,...
Answer:
3n + 1
Step-by-step explanation:
Note the sequence has a common difference between consecutive terms, that is
7 - 4 = 10 - 7 = 13 - 10 = 16 - 13 = 3
This indicates that the sequence is arithmetic with n th term
[tex]a_{n}[/tex] = a₁ + (n - 1)d
where a₁ is the first term and d the common difference
here a₁ = 4 and d = 3, thus
[tex]a_{n}[/tex] = 4 + 3(n - 1) = 4 + 3n - 3 = 3n + 1
Answer:
[tex]\large\boxed{a(n)=3n+1}[/tex]
Step-by-step explanation:
It's an arithmetic sequence:
[tex]a(1)=4\\\\a(2)=a(1)+3=4+3=7\\\\a(3)=a(2)+3=7+3=10\\\\a(4)=a(3)+3=10+3=13\\\\a(5)=a(4)+3=13+3=16\\\vdots[/tex]
An explicit formula of an arithmetic sequence:
[tex]a(n)=a(1)+(n-1)d[/tex]
d - common difference
We have
[tex]a(1)=4,\ d=3[/tex]
Substitute:
[tex]a(n)=4+(n-1)(3)[/tex] use the distributive property
[tex]a(n)=4+3n-3[/tex]
[tex]a(n)=3n+1[/tex]
In large cities people out and take taxis to get from one place to another the cost is $25.20 every 36 miles how much is a 47 mile taxi ride
47 miles taxi ride will cost $32.9
Step-by-step explanation:
Given,
Cost every 36 miles = $25.20
Cost of one mile = [tex]\frac{Cost\ for\ 36\ miles}{36}[/tex]
Cost of one mile = [tex]\frac{25.20}{36}\\[/tex]
Cost of one mile = $0.7
Cost of 47 miles = Cost of one mile * 47
[tex]Cost\ of\ 47\ miles = 0.7*47\\Cost\ of\ 47\ miles = \$32.9[/tex]
47 miles taxi ride will cost $32.9
Keywords: Division, multiplication
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What equation is the result of solving 7+4x=3y for x
Answer:
x=3y/4 - 7/4
Step-by-step explanation:
Answer:
x = [tex]\frac{3y-7}{4}[/tex]
Step-by-step explanation:
Given
7 + 4x = 3y ( subtract 7 from both sides )
4x = 3y - 7 ( divide both sides by 4 )
x = [tex]\frac{3y-7}{4}[/tex]
A painter can paint a wall with an area of 3280 square feet with 8 gallons of paint which rate best represents the relationship between the area of the wall and the amount of paint needed
Answer:
410:1
Step-by-step explanation:
simplify the following (y^2 -3y+4) (5y-3)
Answer:
y = 7/8 = 0.875
Step-by-step explanation:
hope it helps ^^
Find the measure of the supplement of a 95 degree angle
Answer: it is 85 degrees
Step-by-step explanation:
Write an equivalent expression for 48+18x
Answer:
2(24+9x)
Step-by-step explanation:
Answer:
6(8+3x)
Step-by-step explanation:
48+18x=6(8+3x)
20 points for this question:
Find the domain.
Answer:
The Domain D = {y ∈ R / y ≠ [tex]\dfrac{b}{2}[/tex] y ≠ 0}
Step-by-step explanation:
Let ,
f(y) = [tex]\dfrac{b^{2} - 4by }{2y^{2} - by }[/tex] - [tex]\dfrac{4y}{b-2y}[/tex]
The above function would be defined only when the denominator of anyone or both the terms of the function do not become 0 -
[tex]2y^{2} - by[/tex] ≠ 0 and b - 2y ≠ 0
∴ y(2y - b) ≠ 0 and y ≠[tex]\dfrac{b}{2}[/tex]
Either y ≠ 0 or y ≠ [tex]\dfrac{b}{2}[/tex]
∵ The domain of a function consists of only those points where the function is defined -
The Domain D consists of all y ∈ Real numbers such that y ≠ 0 and y ≠ [tex]\dfrac{b}{2}[/tex]
Trapezoid G H J K is rotated about G 90 degrees counterclockwise to form trapezoid G prime H prime J prime K prime. Trapezoid G prime H prime J prime K prime is reflected across the line of reflection m to form trapezoid G double-prime H double-prime J double-prime K double-prime.
What is the final transformation in the composition of transformations that maps pre-image GHJK to image G"H"J"K"?
a translation to the right
a reflection across line m
a 90° rotation about point G
a 180° rotation about point G
Answer:
c
Step-by-step explanation:
Answer:
D
Step-by-step explanation:
on edge
20% of the students in the class play an instrument. If there are 35 students in class, how many play an instrument?
Answer: 7
Step-by-step explanation:
35 x 0.20 = 7
The length of the long leg of a right triangle is 16 inches more than four times of the shorter leg. The length of the hypotenuse is 17 inches more than four times the length of the shorter leg. Find the side lengths of the triangle.
Answer:
The side lengths of the triangle, in order, are 11 inches, 60 inches, 61 inches.
Step-by-step explanation:
Let the shorter leg be x, the longer leg, y and the hypotenus, z.
y = 16 + 4x
z = 17 + 4x
By Pythagoras rule,
z^2 = x^2 + y^2
(17+4x)^2 = x^2 + (16+4x)^2
that is,
x^2 - 8x - 33 = 0 (A quadratic equation)
By factorising,
(x+3)(x-11)=0
either x = -3 or x = 11.
But length cannot be negative. Hence, -3 is dicarded.
x = 11 inches
y = 16 + 4x = 60 inches
z = 17 + 4x = 61 inches
Final answer:
To find the side lengths of the triangle, assume the shorter leg as 'x' inches. Use the given information about the longer leg and hypotenuse to write equations. Simplify the equation and solve for 'x' to find the side lengths of the triangle.
Explanation:
To find the side lengths of the triangle, let's assume that the shorter leg of the right triangle is 'x' inches.
According to the problem, the longer leg is 16 inches more than four times the shorter leg, so it can be written as '4x + 16' inches.
The hypotenuse is 17 inches more than four times the length of the shorter leg, so it can be written as '4x + 17' inches.
Applying the Pythagorean theorem (a² + b² = c²) to the triangle, we get (x² + (4x + 16)² = (4x + 17)²).
Simplifying the equation and solving for 'x', we find that the shorter leg is 15 inches, the longer leg is 76 inches, and the hypotenuse is 77 inches.
1)
► If the ratio of cars to trucks in a parking lot is 8 to 3 and there are 21 trucks, how many cars are there? I don’t know where to start I need help!! This needs to be shown full work on my college math class that’s a requirement!! Can anyone give me an explanation ASAP!!!
Answer:
So a ratio of 8:3 means for every 8 cars that is 3 trucks so we have the number of trucks which is 21 so what i did to find the answer was see how many times 3 went into 21 which is 7 times then you would take the 7 and times it by 8 and get 56 so there are 56 cars.
Step-by-step explanation:
Final answer:
To determine the number of cars, we use the ratio of cars to trucks (8 to 3) with the total number of trucks (21) to find the common multiplier. Multiplying the ratio number for cars (8) by the multiplier (7) gives us 56 cars in the parking lot.
Explanation:
Calculating the Number of Cars in a Parking Lot
To calculate the number of cars in the parking lot when we know the ratio of cars to trucks and the number of trucks, we can set up a proportion. Given that the ratio of cars to trucks is 8 to 3 and there are 21 trucks, we can represent the number of cars as 8x and the number of trucks as 3x, where x is the common multiplier for the ratio. Since we know there are 21 trucks, we can set 3x equal to 21 and solve for x:
3x = 21
Divide both sides of the equation by 3:
x = 21 / 3
x = 7
Now that we have the value of x, we can find the number of cars by multiplying the ratio number for cars (8) by x (7):
Number of cars = 8 x 7
Number of cars = 56
Therefore, there are 56 cars in the parking lot.
what is the force of an object with the mass of 27 kg and an acceleration of 4.75m/s?
Answer:
F = 128.25 Newtons
Step-by-step explanation:
The force of an object is the combination of its mass and its acceleration. The formula is:
F = ma
Where
F is the force in Newtons
m is the mass in (kilograms)
a is the acceleration (in m/s^2)
Given,
m = 27
a = 4.75
We now need to find the force, so we plug the known information and solve easily:
F = ma
F = 27 * 4.75
F = 128.25 Newtons
An astronaut weighs 49 kg on earth. On the moon, an
object's weight is 1/6 of its weight on earth. On Mars, an
object's weight is 0.38 of its weight on earth. What is the
difference between the astronaut's weight on Mars and her
weight on the moon? Round your answer to the nearest
tenth. *
approximately 10.4 kg
approximately 1.4 kg
approximately 104 kg
approximately 49 kg
Answer:
The correct answer is A. Approximately 10.4 kg.
Step-by-step explanation:
1. Let's review the information given to us for solving the question:
Weight of the astronaut on Earth = 49 kg
Weight of the astronaut on the moon = 1/6 * 49 kg
Weight of the astronaut on Mars = 0.38 * 49 kg
2. What is the difference between the astronaut's weight on Mars and her weight on the moon? Round your answer to the nearest tenth.
Weight of the astronaut on the moon = 1/6 * 49 kg
Weight of the astronaut on the moon = 8.2 kg
Weight of the astronaut on Mars = 0.38 * 49 kg
Weight of the astronaut on Mars = 18.6 kg
Difference between the astronaut's weight on Mars and her weight on the moon = 18.6 - 8.2
Difference between the astronaut's weight on Mars and her weight on the moon = 10.4 kg
The cost
y
y
(in dollars) to rent a kayak is proportional to the number
x
x
of hours that you rent the kayak. It costs $27 to rent the kayak for 3 hours.
a. Write an equation that represents the situation.
An equation that represents the situation is
y=
y=
.
Question 2
b. Which is the best Interpretation of the slope?
It costs $9 every 3 hours to rent a kayak.
It costs $9 per hour to rent a kayak.
It costs $81 per hour to rent a kayak.
It costs $24 per hour to rent a kayak.
Question 3
c. How much does it cost to rent the kayak for 5 hours?
It costs $ ? to rent the kayak for 5 hours.
a. An equation that represents the situation is y = 9 x
b. The best Interpretation of the slope is " It costs $9 per hour to rent a kayak " ⇒ 2nd answer
c. It costs $45 to rent the kayak for 5 hours
Step-by-step explanation:
Proportional relationship means two quantities vary with each other
If y varies directly as x, then
y ∝ x y = k xk is called the constant of proportionality You can find k by using the initial values of x and yThe cost " y " in (dollars) to rent a kayak is proportional to the
number of hours " x " that you rent the kayak
∵ y ∝ x
∴ y = k x , where k is the constant of proportionality
∵ It costs $27 to rent the kayak for 3 hours
∴ y = 27 and x = 3
- Substitute these value in the equation above
∴ 27 = k (3)
- Divide both sides by 3
∴ 9 = k
∴ y = 9 x
a. An equation that represents the situation is y = 9 x
∵ The slope is the unit rate of the cost
∵ The slope = 9
∴ The unit rate is $9 per hour
b. The best Interpretation of the slope is " It costs $9 per hour to rent a kayak "
∵ y = 9 x
∵ x = 5 hours
- Substitute x by 5 in the equation above
∴ y = 9 (5) = $45
c. It costs $45 to rent the kayak for 5 hours
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Final answer:
The equation representing the situation is y = 9x. The slope indicates that it costs $9 per hour to rent the kayak. For 5 hours of rental, the cost is $45.
Explanation:
If the cost (y) to rent a kayak is proportional to the number (x) of hours the kayak is rented, and it costs $27 to rent the kayak for 3 hours, we can use this information to write a linear equation that represents the situation. The cost is $9 per hour since $27 ÷ 3 hours = $9 per hour. Our equation is thus y = 9x.
For the interpretation of the slope, since we've determined the cost is $9 per hour, the correct interpretation of the slope is that it costs $9 per hour to rent the kayak.
To find out how much it costs to rent the kayak for 5 hours, we simply plug in x = 5 into our equation y = 9x. So, it costs y = 9 × 5 = $45 to rent the kayak for 5 hours.
A local library bought 54 books. Some cost $32 each and some cost $44 each. The total cost of the books was $1,968. How many of the $32 books were purchased?
Answer:
Step-by-step explanation:
I think the answer is 34
To find the number of $32 books purchased, we can set up a system of equations. Solving the system using substitution, we find that the library purchased 34 of the $32 books.
Explanation:To solve this problem, we can set up a system of equations. Let's call the number of $32 books x and the number of $44 books y. We know that x + y = 54, since the library bought a total of 54 books. We also know that the total cost of the books is $1,968, which can be expressed as 32x + 44y = 1968. Now, we can solve this system of equations to find the value of x.
One way to solve this system is by substitution. Solving the first equation for x, we get x = 54 - y. Plugging this into the second equation, we have 32(54 - y) + 44y = 1968.
Expanding and simplifying, we have 1728 - 32y + 44y = 1968. Combining like terms, we get 12y = 240. Dividing both sides by 12, we find that y = 20. Plugging this back into the first equation x + 20 = 54, we get x = 34.
Therefore, the library purchased 34 of the $32 books.
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