pierdoodsu

2021-12-31

To find: the value of x in proportion
a) $\frac{x}{6}=\frac{3}{x}$
b) $\frac{x-5}{3}=\frac{2x-3}{7}$
c) $\frac{6}{x+4}=\frac{2}{x+2}$
d) $\frac{x+3}{5}=\frac{x+5}{7}$
e) $\frac{x-2}{x-5}=\frac{2x+1}{x-1}$
f) $\frac{x\left(x+5\right)}{4x+4}=\frac{9}{5}$
g) $\frac{x+7}{2}=\frac{x+2}{x-2}$

sirpsta3u

Step 1
a) By using mean-extremes property,
$\frac{x}{6}=\frac{3}{x}$
${x}^{2}=18$
$x=\sqrt{18}$
$x=3\sqrt{2}$
The value of x is $3\sqrt{2}$
b) Then,
$\frac{x-5}{3}=\frac{2x-3}{7}$
$7\left(x-5\right)=3\left(2x-3\right)$
$7x-35=6x-9$
$7x-6x=35-9$
$x=26$
The value of x is 26
c) Given: $\frac{6}{x+4}=\frac{2}{x+2}$
$6\left(x+2\right)=2\left(x+4\right)$
$6x+12=2x+8$
$6x-2x=8-12$
$4x=-4$
$x=-1$
Step 2
d) $\frac{x+3}{5}=\frac{x+5}{7}$
$7\left(x+3\right)=5\left(x+5\right)$
$7x+21=5x+25$
$7x-5x=25-21$
$2x=4$
$x=2$
e) $\frac{x-2}{x-5}=\frac{2x+1}{x-1}$
$\left(x-2\right)\left(x-1\right)=\left(x-5\right)\left(2x+1\right)$
${x}^{2}-x-2x+2=2{x}^{2}+x-10x-5$
${x}^{2}-3x+2=2{x}^{2}-9x-5$
$-{x}^{2}+6x+7=0$
${x}^{2}-6x-7=0$
${x}^{2}-7x+1x+7=0$
$\left(x+1\right)\left(x-7\right)=0$
$x=7$ or $x=-1$
f) Given: $\frac{x\left(x+5\right)}{4x+4}=\frac{9}{5}$

Janet Young

Step 1
a) $\frac{x}{6}=\frac{3}{x}$
Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 6x, the least common multiple of 6, x.
$x\cdot x=6\cdot 3$
Multiply x and x to get ${x}^{2}$
${x}^{2}=6\cdot 3$
Multiply 6 and 3 to get 18.
${x}^{2}$
Take the square root of both sides of the equation.
$x=3\sqrt{2}$
$x=-3\sqrt{2}$
b) $\frac{x-5}{3}=\frac{2x-3}{7}$
Apply fraction cross multiply: if $\frac{a}{b}=\frac{c}{d}$ then $a\cdot d=b\cdot c$
$\left(x-5\right)\cdot 7=3\left(2x-3\right)$
Expand $\left(x-5\right)\cdot 7:$
$7x-35$
Expand $3\left(2x-3\right):$
$6x-9$
$7x-35=6x-9$
$7x-35+35=6x-9+35$
Simplify
$7x=6x+26$
Subtract 6x from both sides
$7x-6x=6x+26-6x$
Simplify
$x=26$
Step 2
c) $\frac{6}{x+4}=\frac{2}{x+2}$
Variable x cannot be equal to any of the values -4, -2 since division by zero is not defined. Multiply both sides of the equation by $\left(x+2\right)\left(x+4\right)$, the least common multiple of
$\left(x+2\right)\cdot 6=\left(x+4\right)\cdot 2$
Use the distributive property to multiply $x+2$ by 6.
$6x+12=\left(x+4\right)\cdot 2$
Use the distributive property to multiply $x+4$ by 2.
$6x+12=2x+8$
Subtract 2x from both sides.
$6x+12-2x=8$
Combine 6x and -2x to get 4x.
$4x+12=8$
Subtract 12 from both sides.
$4x=8-12$
Subtract 12 from 8 to get -4.
$4x=-4$
Divide both sides by 4.
$x=\frac{-4}{4}$
Divide -4 by 4 to get -1.
$x=-1$

karton

Step 1
d) $\frac{x+3}{5}=\frac{x+5}{7}$
Multiply both sides of the equation by 35, the least common multiple of 5, 7.
$7\left(x+3\right)=5\left(x+5\right)$
Use the distributive property to multiply 7 by x+3.
$7x+21=5\left(x+5\right)$
Use the distributive property to multiply 5 by x+5.
$7x+21=5x+25$
Subtract 5x from both sides.
$7x+21-5x=25$
Combine 7x and -5x to get 2x.
$2x+21=25$
Subtract 21 from both sides.
$2x=25-21$
Subtract 21 from 25 to get 4.
$2x=4$
Divide both sides by 2.
$x=\frac{4}{2}$
Divide 4 by 2 to get 2.
x=2
Step 2
e) $\frac{x-2}{x-5}=\frac{2x+1}{x-1}$
Apply fraction cross multiply: if $\frac{a}{b}=\frac{c}{d}$ then $a×d=b×c$
$\left(x-2\right)\left(x-1\right)=\left(x-5\right)\left(2x+1\right)$
Solve (x-2)(x-1)=(x-5)(2x+1):
x=-1, x=7
Verify Solutions
Find undefined (signularity) points: x=5, x=1
Step 3
f) $\frac{x\left(x+5\right)}{4x+4}=\frac{9}{5}$
NSK
Variable x cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by 20(x+1), the least common multiple of 4x+4,\ 5.
$5x\left(x+5\right)=36\left(x+1\right)$
Use the distributive property to multiply 5x by x+5.
$5{x}^{2}+25x=36\left(x+1\right)$
Use the distributive property to multiply 36 by x+1.
$5{x}^{2}+25x=36x+36$
Subtract 36x from both sides.
$5{x}^{2}+25x-36x=36$
Combine 25x and -36x to get -11x.
$5{x}^{2}-11x=36$
Subtract 36 from both sides.
$5{x}^{2}-11x-36=0$
This equation is in standard form: $a{x}^{2}+bx+c=0$
Substitute 5 for a, -11 for b, and -36 for c in the quadratic formula $-b±\sqrt{{b}^{2}-4ac}2a$
$x=\frac{-\left(-11\right)±\sqrt{\left(-11{\right)}^{2}-4×5\left(-36\right)}}{2×5}$
Square -11
$x=\frac{-\left(-11\right)±\sqrt{121-4×5\left(-36\right)}}{2×5}$
Multiply -4 times 5
$x=\frac{-\left(-11\right)±\sqrt{121-20\left(-36\right)}}{2×5}$
Multiply -20 times -36.
$x=\frac{-\left(-11\right)±\sqrt{121+720}}{2×5}$
$x=\frac{-\left(-11\right)±\right]\sqrt{841}}{2×5}$
Take the square root of 841.
$x=\frac{-\left(-11\right)±29}{2×5}$
The opposite of -11 is 11.
$x=\frac{11±29}{2×5}$
Multiply 2 times 5.
$x=\frac{11±29}{10}$
Now solve the equation $x=\frac{11±29}{10}$ when $±$ is plus. Add 11 to 29
$x=\frac{40}{10}$
Divide 40 by 10.
x=4
Now solve the equation $x=\frac{11±29}{10}$ when $±$ is minus. Subtract 29 from 11
$x=\frac{-18}{10}$
Reduce the fraction $\frac{-18}{10}$ to lowest terms by extracting and canceling out 2.
$x=-\frac{9}{5}$
The equation is now solved.
$x=4$
$x=-\frac{9}{5}$
Step 4
g) $\frac{x+7}{2}=\frac{x+2}{x-2}$
Variable x cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by 2(x-2), the least common multiple of 2, x-2.
$\left(x-2\right)\left(x+7\right)=2\left(x+2\right)$
Use the distributive property to multiply x-2 by x+7 and combine like terms.
${x}^{2}+5x-14=2\left(x+2\right)$
Use the distributive property to multiply 2 by x+2.
${x}^{2}+5x-14=2x+4$
Subtract 2x from both sides.
${x}^{2}+5x-14-2x=4$
Combine 5x and -2x to get 3x.

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