directions: solve each proportion
13. $\frac{9}{18}=\frac{x}{12}$
$\frac{18}{2}=\frac{2x}{2}$
$14 = x$
14. $\frac{x - 3}{18}=\frac{12}{9}$
$14\cdot12 = 10x$
$\frac{228}{10}=\frac{10x}{10}$
$x = 22.8$
15. $\frac{7}{11}=\frac{18}{x + 1}$
16. $\frac{3x - 4}{14}=\frac{9}{10}$
17. $\frac{17}{15}=\frac{10}{2x - 2}$
18. $\frac{x - 16}{x + 6}=\frac{3}{5}$
19. $\frac{6}{19}=\frac{x - 12}{2x - 2}$
20. $\frac{x - 9}{15}=\frac{2x - 9}{10}$
21. $\frac{x - 9}{3}=\frac{56}{x + 4}$
22. $\frac{7}{x + 1}=\frac{2x - 1}{36}$

Question

Accepted Answer

Let's solve each proportion one by one:

### Problem 13: $\boldsymbol{\frac{9}{18} = \frac{x}{12}}$
# Explanation:
## Step 1: Cross - multiply
Cross - multiplying gives us $9	imes12 = 18	imes x$.
So, $108=18x$.
## Step 2: Solve for $x$
Divide both sides by 18: $x=\frac{108}{18}=6$.
# Answer:
$x = 6$

### Problem 14: $\boldsymbol{\frac{x - 3}{18}=\frac{12}{9}}$
# Explanation:
## Step 1: Cross - multiply
Cross - multiplying gives $(x - 3)	imes9=18	imes12$.
$9x-27 = 216$.
## Step 2: Add 27 to both sides
$9x=216 + 27=243$.
## Step 3: Solve for $x$
Divide both sides by 9: $x=\frac{243}{9}=27$.
# Answer:
$x = 27$

### Problem 15: $\boldsymbol{\frac{7}{11}=\frac{18}{x + 1}}$
# Explanation:
## Step 1: Cross - multiply
Cross - multiplying gives $7	imes(x + 1)=11	imes18$.
$7x+7 = 198$.
## Step 2: Subtract 7 from both sides
$7x=198 - 7 = 191$.
## Step 3: Solve for $x$
$x=\frac{191}{7}\approx27.29$ (or as an improper fraction $\frac{191}{7}$)
# Answer:
$x=\frac{191}{7}$ (or $x\approx27.29$)

### Problem 16: $\boldsymbol{\frac{3x - 4}{14}=\frac{9}{10}}$
# Explanation:
## Step 1: Cross - multiply
Cross - multiplying gives $(3x - 4)	imes10=14	imes9$.
$30x-40 = 126$.
## Step 2: Add 40 to both sides
$30x=126 + 40=166$.
## Step 3: Solve for $x$
$x=\frac{166}{30}=\frac{83}{15}\approx5.53$
# Answer:
$x=\frac{83}{15}$ (or $x\approx5.53$)

### Problem 17: $\boldsymbol{\frac{17}{15}=\frac{10}{2x - 2}}$
# Explanation:
## Step 1: Cross - multiply
Cross - multiplying gives $17	imes(2x - 2)=15	imes10$.
$34x-34 = 150$.
## Step 2: Add 34 to both sides
$34x=150 + 34 = 184$.
## Step 3: Solve for $x$
$x=\frac{184}{34}=\frac{92}{17}\approx5.41$
# Answer:
$x=\frac{92}{17}$ (or $x\approx5.41$)

### Problem 18: $\boldsymbol{\frac{x - 16}{x + 6}=\frac{3}{5}}$
# Explanation:
## Step 1: Cross - multiply
Cross - multiplying gives $5	imes(x - 16)=3	imes(x + 6)$.
$5x-80 = 3x + 18$.
## Step 2: Subtract $3x$ from both sides
$5x-3x-80=3x-3x + 18$.
$2x-80 = 18$.
## Step 3: Add 80 to both sides
$2x=18 + 80=98$.
## Step 4: Solve for $x$
$x=\frac{98}{2}=49$.
# Answer:
$x = 49$

### Problem 19: $\boldsymbol{\frac{6}{19}=\frac{x - 12}{2x - 2}}$
# Explanation:
## Step 1: Cross - multiply
Cross - multiplying gives $6	imes(2x - 2)=19	imes(x - 12)$.
$12x-12 = 19x-228$.
## Step 2: Subtract $12x$ from both sides
$12x-12x-12=19x-12x - 228$.
$- 12=7x-228$.
## Step 3: Add 228 to both sides
$-12 + 228=7x-228 + 228$.
$216 = 7x$.
## Step 4: Solve for $x$
$x=\frac{216}{7}\approx30.86$ (or as an improper fraction $\frac{216}{7}$)
# Answer:
$x=\frac{216}{7}$ (or $x\approx30.86$)

### Problem 20: $\boldsymbol{\frac{x - 9}{15}=\frac{2x - 9}{10}}$
# Explanation:
## Step 1: Cross - multiply
Cross - multiplying gives $10	imes(x - 9)=15	imes(2x - 9)$.
$10x-90 = 30x-135$.
## Step 2: Subtract $10x$ from both sides
$10x-10x-90=30x-10x - 135$.
$-90 = 20x-135$.
## Step 3: Add 135 to both sides
$-90 + 135=20x-135 + 135$.
$45 = 20x$.
## Step 4: Solve for $x$
$x=\frac{45}{20}=\frac{9}{4}=2.25$.
# Answer:
$x=\frac{9}{4}$ (or $x = 2.25$)

### Problem 21: $\boldsymbol{\frac{x - 9}{3}=\frac{56}{x + 4}}$
# Explanation:
## Step 1: Cross - multiply
Cross - multiplying gives $(x - 9)	imes(x + 4)=3	imes56$.
$x^{2}+4x-9x - 36=168$.
$x^{2}-5x-36 - 168=0$.
$x^{2}-5x-204 = 0$.
## Step 2: Factor the quadratic equation
We need to find two numbers that multiply to $- 204$ and add up to $-5$. The numbers are $-17$ and $12$.
So, $x^{2}-17x + 12x-204 = 0$.
$x(x - 17)+12(x - 17)=0$.
$(x - 17)(x + 12)=0$.
## Step 3: Solve for $x$
Setting each factor equal to zero gives $x-17 = 0$ or $x + 12=0$.
So, $x = 17$ or $x=-12$.
We need to check for extraneous solutions.
If $x=-12$, the denominator $x + 4=-12 + 4=-8
eq0$ and the denominator of the left - hand side $3
eq0$. But when we substitute $x=-12$ into the original equation:
Left - hand side: $\frac{-12 - 9}{3}=\frac{-21}{3}=-7$
Right - hand side: $\frac{56}{-12 + 4}=\frac{56}{-8}=-7$. So $x=-12$ is a valid solution.
If $x = 17$, left - hand side: $\frac{17-9}{3}=\frac{8}{3}$, right - hand side: $\frac{56}{17 + 4}=\frac{56}{21}=\frac{8}{3}$. So $x = 17$ is also a valid solution.
# Answer:
$x = 17$ or $x=-12$

### Problem 22: $\boldsymbol{\frac{7}{x + 1}=\frac{2x - 1}{36}}$
# Explanation:
## Step 1: Cross - multiply
Cross - multiplying gives $7	imes36=(x + 1)	imes(2x - 1)$.
$252=2x^{2}-x + 2x-1$.
$2x^{2}+x-1 - 252=0$.
$2x^{2}+x-253 = 0$.
## Step 2: Use the quadratic formula
For a quadratic equation $ax^{2}+bx + c = 0$, $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. Here, $a = 2$, $b = 1$, $c=-253$.
First, calculate the discriminant $\Delta=b^{2}-4ac=(1)^{2}-4	imes2	imes(-253)=1 + 2024 = 2025$.
Then, $x=\frac{-1\pm\sqrt{2025}}{4}=\frac{-1\pm45}{4}$.
We have two solutions:
$x_{1}=\frac{-1 + 45}{4}=\frac{44}{4}=11$ and $x_{2}=\frac{-1-45}{4}=\frac{-46}{4}=-\frac{23}{2}=-11.5$.
Check for extraneous solutions:
For $x = 11$:
Left - hand side: $\frac{7}{11 + 1}=\frac{7}{12}$, Right - hand side: $\frac{2	imes11-1}{36}=\frac{21}{36}=\frac{7}{12}$. Valid.
For $x=-\frac{23}{2}$:
Left - hand side: $\frac{7}{-\frac{23}{2}+1}=\frac{7}{-\frac{21}{2}}=-\frac{2}{3}$, Right - hand side: $\frac{2	imes(-\frac{23}{2})-1}{36}=\frac{-23 - 1}{36}=\frac{-24}{36}=-\frac{2}{3}$. Valid.
# Answer:
$x = 11$ or $x=-\frac{23}{2}$

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QUESTION IMAGE

Question

Explanation:

Problem 13: $\boldsymbol{\frac{9}{18} = \frac{x}{12}}$

Step 1: Cross - multiply

Step 2: Solve for $x$

Step 1: Cross - multiply

Step 2: Add 27 to both sides

Step 3: Solve for $x$

Step 1: Cross - multiply

Step 2: Subtract 7 from both sides

Step 3: Solve for $x$

Answer:

Problem 14: $\boldsymbol{\frac{x - 3}{18}=\frac{12}{9}}$