Math / Algebra

QuestionSay $z$ varies directly as $x$ and inversely as $y$ . $z$ is $\frac{18}{5}$ when $x$ is 6 and $y$ is 5 . What is $x$ when $y$ is 9 and $z$ is 3 ?

Studdy Solution

STEP 1

What is this asking? If $z$ changes based on $x$ and $y$ in a specific way, and we know one set of values for $x$ , $y$ , and $z$ , can we figure out $x$ given different values for $y$ and $z$ ? Watch out! Don't mix up direct and inverse variation!
Direct means they change in the same way, inverse means they change in opposite ways.

STEP 2

1. Set up the variation equation.
2. Find the constant of variation.
3. Solve for x.

STEP 3

Alright, so we know that $z$ varies directly with $x$ and inversely with $y$ .
This tells us we can set up an equation like this: $z = k \cdot \frac{x}{y}$ , where $k$ is our constant of variation.
We use $k$ because the relationship between $z$ , $x$ , and $y$ always stays the same.

STEP 4

We're given that $z$ is $\frac{18}{5}$ when $x$ is 6 and $y$ is 5.
Let's plug these values into our equation: $\frac{18}{5} = k \cdot \frac{6}{5}$ .

STEP 5

Now, to solve for $k$ , we can multiply both sides of the equation by $\frac{5}{6}$ .
This is because $\frac{5}{6} \cdot \frac{6}{5} = 1$ , and multiplying by one doesn't change the value!
So, we get $\frac{18}{5} \cdot \frac{5}{6} = k$ .

STEP 6

Let’s simplify!
Notice that we can divide to one!
We have a 5 in the numerator and denominator of the left side, so $\frac{5}{5} = 1$ .
Also, 6 goes into 18 three times, so $\frac{18}{6} = 3$ .
This gives us $3 \cdot 1 = k$ , or $k = 3$ !
Our constant of variation is 3!

STEP 7

Now we know our equation is $z = 3 \cdot \frac{x}{y}$ .
We want to find $x$ when $y$ is 9 and $z$ is 3.
Let's plug in those values: $3 = 3 \cdot \frac{x}{9}$ .

STEP 8

To get $x$ by itself, we can first divide both sides by 3.
Since $\frac{3}{3} = 1$ , we have $1 = \frac{x}{9}$ .

STEP 9

Finally, multiply both sides by 9, because $9 \cdot \frac{1}{9} = 1$ .
This gives us $9 = x$ , or $x = 9$ !

STEP 10

When $y$ is 9 and $z$ is 3, $x$ is 9!

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QuestionSay $z$ varies directly as $x$ and inversely as $y$ . $z$ is $\frac{18}{5}$ when $x$ is 6 and $y$ is 5 . What is $x$ when $y$ is 9 and $z$ is 3 ?

Studdy Solution

STEP 1

STEP 2

1. Set up the variation equation.
2. Find the constant of variation.
3. Solve for x.

STEP 3

Alright, so we know that $z$ varies directly with $x$ and inversely with $y$ .
This tells us we can set up an equation like this: $z = k \cdot \frac{x}{y}$ , where $k$ is our constant of variation.
We use $k$ because the relationship between $z$ , $x$ , and $y$ always stays the same.

STEP 4

We're given that $z$ is $\frac{18}{5}$ when $x$ is 6 and $y$ is 5.
Let's plug these values into our equation: $\frac{18}{5} = k \cdot \frac{6}{5}$ .

STEP 5

Now, to solve for $k$ , we can multiply both sides of the equation by $\frac{5}{6}$ .
This is because $\frac{5}{6} \cdot \frac{6}{5} = 1$ , and multiplying by one doesn't change the value!
So, we get $\frac{18}{5} \cdot \frac{5}{6} = k$ .

STEP 6

Let’s simplify!
Notice that we can divide to one!
We have a 5 in the numerator and denominator of the left side, so $\frac{5}{5} = 1$ .
Also, 6 goes into 18 three times, so $\frac{18}{6} = 3$ .
This gives us $3 \cdot 1 = k$ , or $k = 3$ !
Our constant of variation is 3!

STEP 7

Now we know our equation is $z = 3 \cdot \frac{x}{y}$ .
We want to find $x$ when $y$ is 9 and $z$ is 3.
Let's plug in those values: $3 = 3 \cdot \frac{x}{9}$ .

STEP 8

To get $x$ by itself, we can first divide both sides by 3.
Since $\frac{3}{3} = 1$ , we have $1 = \frac{x}{9}$ .

STEP 9

Finally, multiply both sides by 9, because $9 \cdot \frac{1}{9} = 1$ .
This gives us $9 = x$ , or $x = 9$ !

STEP 10

When $y$ is 9 and $z$ is 3, $x$ is 9!

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QuestionSay zzz varies directly as xxx and inversely as yyy. zzz is 185\frac{18}{5}518​ when xxx is 6 and yyy is 5 . What is xxx when yyy is 9 and zzz is 3 ?

Studdy Solution

STEP 1

STEP 2

1. Set up the variation equation.2. Find the constant of variation.3. Solve for *x*.

STEP 3

STEP 4

We're given that zzz is 185\frac{18}{5}518​ when xxx is 6 and yyy is 5.Let's plug these **values** into our equation: 185=k⋅65\frac{18}{5} = k \cdot \frac{6}{5}518​=k⋅56​.

STEP 5

Now, to solve for kkk, we can multiply both sides of the equation by 56\frac{5}{6}65​.This is because 56⋅65=1\frac{5}{6} \cdot \frac{6}{5} = 165​⋅56​=1, and multiplying by one doesn't change the value!So, we get 185⋅56=k\frac{18}{5} \cdot \frac{5}{6} = k518​⋅65​=k.

STEP 6

STEP 7

Now we know our equation is z=3⋅xyz = 3 \cdot \frac{x}{y}z=3⋅yx​.We want to find xxx when yyy is 9 and zzz is 3.Let's plug in those **values**: 3=3⋅x93 = 3 \cdot \frac{x}{9}3=3⋅9x​.

STEP 8

To get xxx by itself, we can first divide both sides by 3.Since 33=1\frac{3}{3} = 133​=1, we have 1=x91 = \frac{x}{9}1=9x​.

STEP 9

Finally, multiply both sides by 9, because 9⋅19=19 \cdot \frac{1}{9} = 19⋅91​=1.This gives us 9=x9 = x9=x, or x=9x = 9x=9!

STEP 10

When yyy is 9 and zzz is 3, xxx is 9!

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QuestionSay $z$ varies directly as $x$ and inversely as $y$ . $z$ is $\frac{18}{5}$ when $x$ is 6 and $y$ is 5 . What is $x$ when $y$ is 9 and $z$ is 3 ?

1. Set up the variation equation.
2. Find the constant of variation.
3. Solve for x.

We're given that $z$ is $\frac{18}{5}$ when $x$ is 6 and $y$ is 5.
Let's plug these values into our equation: $\frac{18}{5} = k \cdot \frac{6}{5}$ .

Now, to solve for $k$ , we can multiply both sides of the equation by $\frac{5}{6}$ .
This is because $\frac{5}{6} \cdot \frac{6}{5} = 1$ , and multiplying by one doesn't change the value!
So, we get $\frac{18}{5} \cdot \frac{5}{6} = k$ .

Now we know our equation is $z = 3 \cdot \frac{x}{y}$ .
We want to find $x$ when $y$ is 9 and $z$ is 3.
Let's plug in those values: $3 = 3 \cdot \frac{x}{9}$ .

To get $x$ by itself, we can first divide both sides by 3.
Since $\frac{3}{3} = 1$ , we have $1 = \frac{x}{9}$ .

Finally, multiply both sides by 9, because $9 \cdot \frac{1}{9} = 1$ .
This gives us $9 = x$ , or $x = 9$ !

When $y$ is 9 and $z$ is 3, $x$ is 9!