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)

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.
$7(x+3)=5(x+5)$
Use the distributive property to multiply 7 by x+3.
$7x+21=5(x+5)$
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 $a\times d=b\times c$
$(x-2)(x-1)=(x-5)(2x+1)$
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) $\frac{x(x+5)}{4x+4}=\frac{9}{5}$
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.
$5x(x+5)=36(x+1)$
Use the distributive property to multiply 5x by x+5.
$5x^2+25x=36(x+1)$
Use the distributive property to multiply 36 by x+1.
$5x^2+25x=36x+36$
Subtract 36x from both sides.
$5x^2+25x-36x=36$
Combine 25x and -36x to get -11x.
$5x^2-11x=36$
Subtract 36 from both sides.
$5x^2-11x-36=0$
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 $-b\pm \sqrt{b^2-4ac}2a$
$x=\frac{-(-11)\pm \sqrt{(-11{)}^2-4\times 5(-36)}}{2\times 5}$
Square -11
$x=\frac{-(-11)\pm \sqrt{121-4\times 5(-36)}}{2\times 5}$
Multiply -4 times 5
$x=\frac{-(-11)\pm \sqrt{121-20(-36)}}{2\times 5}$
Multiply -20 times -36.
$x=\frac{-(-11)\pm \sqrt{121+720}}{2\times 5}$
Add 121 to 720.
$x=\frac{-(-11)\pm ]\sqrt{841}}{2\times 5}$
Take the square root of 841.
$x=\frac{-(-11)\pm 29}{2\times 5}$
The opposite of -11 is 11.
$x=\frac{11\pm 29}{2\times 5}$
Multiply 2 times 5.
$x=\frac{11\pm 29}{10}$
Now solve the equation $x=\frac{11\pm 29}{10}$ when $\pm$ is plus. Add 11 to 29
$x=\frac{40}{10}$
Divide 40 by 10.
x=4
Now solve the equation $x=\frac{11\pm 29}{10}$ when $\pm$ 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.
$(x-2)(x+7)=2(x+2)$
Use the distributive property to multiply x-2 by x+7 and combine like terms.
$x^2+5x-14=2(x+2)$
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.