Practice:  More Linear Equations (2.4, 2.6, 2.7)

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44 Practice: More Linear Equations (2.4, 2.6, 2.7)

2.4 Fractional Linear Equations

For questions 1 to 18, solve each linear equation.

$\dfrac{3}{5}\left(1 + p\right) = \dfrac{21}{20}$
$-\dfrac{1}{2} = \dfrac{3k}{2} + \dfrac{3}{2}$
$0 = -\dfrac{5}{4}\left(x-\dfrac{6}{5}\right)$
$\dfrac{3}{2}n - 8 = -\dfrac{29}{12}$
$\dfrac{3}{4} - \dfrac{5}{4}m = \dfrac{108}{24}$
$\dfrac{11}{4} + \dfrac{3}{4}r = \dfrac{160}{32}$
$2b + \dfrac{9}{5} = -\dfrac{11}{5}$
$\dfrac{3}{2} - \dfrac{7}{4}v = -\dfrac{9}{8}$
$\dfrac{3}{2}\left(\dfrac{7}{3}n+1\right) = \dfrac{3}{2}$
$\dfrac{41}{9} = \dfrac{5}{2}\left(x+\dfrac{2}{3}\right) - \dfrac{1}{3}x$
$-a - \dfrac{5}{4}\left(-\dfrac{8}{3}a+ 1\right) = -\dfrac{19}{4}$
$\dfrac{1}{3}\left(-\dfrac{7}{4}k + 1\right) - \dfrac{10}{3}k = -\dfrac{13}{8}$
$\dfrac{55}{6} = -\dfrac{5}{2}\left(\dfrac{3}{2}p-\dfrac{5}{3}\right)$
$-\dfrac{1}{2}\left(\dfrac{2}{3}x-\dfrac{3}{4}\right)-\dfrac{7}{2}x=-\dfrac{83}{24}$
$-\dfrac{5}{8}=\dfrac{5}{4}\left(r-\dfrac{3}{2}\right)$
$\dfrac{1}{12}=\dfrac{4}{3}x+\dfrac{5}{3}\left(x-\dfrac{7}{4}\right)$
$-\dfrac{11}{3}+\dfrac{3}{2}b=\dfrac{5}{2}\left(b-\dfrac{5}{3}\right)$
$\dfrac{7}{6}-\dfrac{4}{3}n=-\dfrac{3}{2}n+2\left(n+\dfrac{3}{2}\right)$

<a class=”internal” href=”/intermediatealgebraberg/back-matter/answer-key-2-4/”>Answer Key 2.4

2.6 Working With Formulas — Intermediate Algebra

For questions 1 to 10, evaluate each expression using the values given.

$p + 1 + q - m\text{ (}m = 1, p = 3, q = 4)$
$y^2+y-z\text{ (}y=5, z=1)$
$p- \left[pq \div 6\right]\text{ (}p=6, q=5)$
$\left[6+z-y\right]\div 3\text{ (}y=1, z=4)$
$c^2-(a-1)\text{ (}a=3, c=5)$
$x+6z-4y\text{ (}x=6, y=4, z=4)$
$5j+kh\div 2\text{ (}h=5, j=4, k=2)$
$5(b+a)+1+c\text{ (}a=2, b=6, c=5)$
$\left[4-(p-m)\right]\div 2+q\text{ (}m=4, p=6, q=6)$
$z+x-(1^2)^3\text{ (}x=5, z=4)$

For questions 11 to 34, isolate the indicated variable from the equation.

$b\text{ in }ab=c$
$h\text{ in }g=\dfrac{h}{i}$
$x\text{ in }\left(\dfrac{f}{g}\right)x=b$
$y\text{ in }p=\dfrac{3y}{q}$
$x\text{ in }3x=\dfrac{a}{b}$
$y\text{ in }\dfrac{ym}{b}=\dfrac{c}{d}$
$\pi\text{ in }V=\dfrac{4}{3}\pi r^3$
$m\text{ in }E=mv^2$
$y\text{ in }c=\dfrac{4y}{m+n}$
$r\text{ in }\dfrac{rs}{a-3}=k$
$D\text{ in }V=\dfrac{\pi Dn}{12}$
$R\text{ in }F=k(R-L)$
$c\text{ in }P=n(p-c)$
$L\text{ in }S=L+2B$
$D\text{ in }T=\dfrac{D-d}{L}$
$E_a\text{ in }I=\dfrac{E_a-E_q}{R}$
$L_o\text{ in }L=L_o(1+at)$
$m\text{ in }2m+p=4m+q$
$k\text{ in }\dfrac{k-m}{r}=q$
$T\text{ in }R=aT+b$
$Q_2\text{ in }Q_1=P(Q_2-Q_1)$
$r_1\text{ in }L=\pi(r_1+r_2)+2d$
$T_1\text{ in }R=\dfrac{kA(T+T_1)}{d}$
$V_2\text{ in }P=\dfrac{V_1(V_2-V_1)}{g}$

<a class=”internal” href=”/intermediatealgebraberg/back-matter/answer-key-2-6/”>Answer Key 2.6

2.7 Variation Word Problems –

For questions 1 to 12, write the formula defining the variation, including the constant of variation $(k).$

$x$ varies directly as $y$
$x$ is jointly proportional to $y$ and $z$
$x$ varies inversely as $y$
$x$ varies directly as the square of $y$
$x$ varies jointly as $z$ and $y$
$x$ is inversely proportional to the cube of $y$
$x$ is jointly proportional with the square of $y$ and the square root of $z$
$x$ is inversely proportional to $y$ to the sixth power
$x$ is jointly proportional with the cube of $y$ and inversely to the square root of $z$
$x$ is inversely proportional with the square of $y$ and the square root of $z$
$x$ varies jointly as $z$ and $y$ and is inversely proportional to the cube of $p$
$x$ is inversely proportional to the cube of $y$ and square of $z$

For questions 13 to 22, find the formula defining the variation and the constant of variation $(k).$

If $A$ varies directly as $B,$ find $k$ when $A=15$ and $B=5.$
If $P$ is jointly proportional to $Q$ and $R,$ find $k$ when $P=12, Q=8$ and $R=3.$
If $A$ varies inversely as $B,$ find $k$ when $A=7$ and $B=4.$
If $A$ varies directly as the square of $B,$ find $k$ when $A=6$ and $B=3.$
If $C$ varies jointly as $A$ and $B,$ find $k$ when $C=24, A=3,$ and $B=2.$
If $Y$ is inversely proportional to the cube of $X,$ find $k$ when $Y=54$ and $X=3.$
If $X$ is directly proportional to $Y,$ find $k$ when $X=12$ and $Y=8.$
If $A$ is jointly proportional with the square of $B$ and the square root of $C,$ find $k$ when $A=25, B=5$ and $C=9.$
If $y$ varies jointly with $m$ and the square of $n$ and inversely with $d,$ find $k$ when $y=10, m=4, n=5,$ and $d=6.$
If $P$ varies directly as $T$ and inversely as $V,$ find $k$ when $P=10, T=250,$ and $V=400.$

For questions 23 to 37, solve each variation word problem.

The electrical current $I$ (in amperes, A) varies directly as the voltage $(V)$ in a simple circuit. If the current is 5 A when the source voltage is 15 V, what is the current when the source voltage is 25 V?
The current $I$ in an electrical conductor varies inversely as the resistance $R$ (in ohms, Ω) of the conductor. If the current is 12 A when the resistance is 240 Ω, what is the current when the resistance is 540 Ω?
Hooke’s law states that the distance $(d_s)$ that a spring is stretched supporting a suspended object varies directly as the mass of the object $(m).$ If the distance stretched is 18 cm when the suspended mass is 3 kg, what is the distance when the suspended mass is 5 kg?
The volume $(V)$ of an ideal gas at a constant temperature varies inversely as the pressure $(P)$ exerted on it. If the volume of a gas is 200 cm³ under a pressure of 32 kg/cm², what will be its volume under a pressure of 40 kg/cm²?
The number of aluminum cans $(c)$ used each year varies directly as the number of people $(p)$ using the cans. If 250 people use 60,000 cans in one year, how many cans are used each year in a city that has a population of 1,000,000?
The time $(t)$ required to do a masonry job varies inversely as the number of bricklayers $(b).$ If it takes 5 hours for 7 bricklayers to build a park wall, how much time should it take 10 bricklayers to complete the same job?
The wavelength of a radio signal (λ) varies inversely as its frequency $(f).$ A wave with a frequency of 1200 kilohertz has a length of 250 metres. What is the wavelength of a radio signal having a frequency of 60 kilohertz?
The number of kilograms of water $(w)$ in a human body is proportional to the mass of the body $(m).$ If a 96 kg person contains 64 kg of water, how many kilograms of water are in a 60 kg person?
The time $(t)$ required to drive a fixed distance $(d)$ varies inversely as the speed $(v).$ If it takes 5 hours at a speed of 80 km/h to drive a fixed distance, what speed is required to do the same trip in 4.2 hours?
The volume $(V)$ of a cone varies jointly as its height $(h)$ and the square of its radius $(r).$ If a cone with a height of 8 centimetres and a radius of 2 centimetres has a volume of 33.5 cm³, what is the volume of a cone with a height of 6 centimetres and a radius of 4 centimetres?
The centripetal force $(F_{\text{c}})$ acting on an object varies as the square of the speed $(v)$ and inversely to the radius $(r)$ of its path. If the centripetal force is 100 N when the object is travelling at 10 m/s in a path or radius of 0.5 m, what is the centripetal force when the object’s speed increases to 25 m/s and the path is now 1.0 m?
The maximum load $(L_{\text{max}})$ that a cylindrical column with a circular cross section can hold varies directly as the fourth power of the diameter $(d)$ and inversely as the square of the height $(h).$ If an 8.0 m column that is 2.0 m in diameter will support 64 tonnes, how many tonnes can be supported by a column 12.0 m high and 3.0 m in diameter?
The volume $(V)$ of gas varies directly as the temperature $(T)$ and inversely as the pressure $(P).$ If the volume is 225 cc when the temperature is 300 K and the pressure is 100 N/cm², what is the volume when the temperature drops to 270 K and the pressure is 150 N/cm²?
The electrical resistance $(R)$ of a wire varies directly as its length $(l)$ and inversely as the square of its diameter $(d).$ A wire with a length of 5.0 m and a diameter of 0.25 cm has a resistance of 20 Ω. Find the electrical resistance in a 10.0 m long wire having twice the diameter.
The volume of wood in a tree $(V)$ varies directly as the height $(h)$ and the diameter $(d).$ If the volume of a tree is 377 m³ when the height is 30 m and the diameter is 2.0 m, what is the height of a tree having a volume of 225 m³ and a diameter of 1.75 m?