Math / Algebra

QuestionSay $z$ varies directly as $x$ and inversely as $y . z$ is $-\frac{9}{2}$ when $x$ is 6 and $y$ is 4 . What is $x$ when $y$ is 6 and $z$ is $-\frac{0}{2}$ ?

Studdy Solution

STEP 1

What is this asking? If $z$ changes proportionally with $x$ and inversely proportionally with $y$ , and we know one set of values for $x$ , $y$ , and $z$ , we need to find $x$ given different values for $y$ and $z$ . Watch out! Don't mix up direct and inverse variation!
Also, be careful with the negative signs!

STEP 2

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

STEP 3

We're told that $z$ varies directly as $x$ and inversely as $y$ .
This translates to $z = \frac{kx}{y}$ , where $k$ is our constant of variation.
We need to find this $k$ before we can do anything else!

STEP 4

We know that $z$ is $-\frac{9}{2}$ when $x$ is 6 and $y$ is 4.
Let's plug these values into our equation: $-\frac{9}{2} = \frac{k \cdot 6}{4}.$

STEP 5

To isolate $k$ , we can multiply both sides of the equation by 4, which gives us: $-\frac{9}{2} \cdot 4 = k \cdot 6.$ $-18 = k \cdot 6.$

STEP 6

Now, divide both sides by 6: $\frac{-18}{6} = k.$ $k = -3.$ So our constant of variation, $k$ , is $-3$ !

STEP 7

Now we know our complete equation: $z = \frac{-3x}{y}$ .
We want to find $x$ when $y$ is 6 and $z$ is $-\frac{0}{2}$ .
Notice that $-\frac{0}{2}$ is just 0!
Let's plug in these values: $0 = \frac{-3x}{6}.$

STEP 8

To solve for $x$ , we can multiply both sides by 6: $0 \cdot 6 = -3x.$ $0 = -3x.$

STEP 9

Finally, divide both sides by $-3$ : $\frac{0}{-3} = x.$ $x = 0.$ There it is!

STEP 10

When $y$ is 6 and $z$ is 0, $x$ is 0.

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QuestionSay $z$ varies directly as $x$ and inversely as $y . z$ is $-\frac{9}{2}$ when $x$ is 6 and $y$ is 4 . What is $x$ when $y$ is 6 and $z$ is $-\frac{0}{2}$ ?

Studdy Solution

STEP 1

STEP 2

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

STEP 3

We're told that $z$ varies directly as $x$ and inversely as $y$ .
This translates to $z = \frac{kx}{y}$ , where $k$ is our constant of variation.
We need to find this $k$ before we can do anything else!

STEP 4

We know that $z$ is $-\frac{9}{2}$ when $x$ is 6 and $y$ is 4.
Let's plug these values into our equation: $-\frac{9}{2} = \frac{k \cdot 6}{4}.$

STEP 5

To isolate $k$ , we can multiply both sides of the equation by 4, which gives us: $-\frac{9}{2} \cdot 4 = k \cdot 6.$ $-18 = k \cdot 6.$

STEP 6

Now, divide both sides by 6: $\frac{-18}{6} = k.$ $k = -3.$ So our constant of variation, $k$ , is $-3$ !

STEP 7

Now we know our complete equation: $z = \frac{-3x}{y}$ .
We want to find $x$ when $y$ is 6 and $z$ is $-\frac{0}{2}$ .
Notice that $-\frac{0}{2}$ is just 0!
Let's plug in these values: $0 = \frac{-3x}{6}.$

STEP 8

To solve for $x$ , we can multiply both sides by 6: $0 \cdot 6 = -3x.$ $0 = -3x.$

STEP 9

Finally, divide both sides by $-3$ : $\frac{0}{-3} = x.$ $x = 0.$ There it is!

STEP 10

When $y$ is 6 and $z$ is 0, $x$ is 0.

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QuestionSay zzz varies directly as xxx and inversely as y.zy . zy.z is −92-\frac{9}{2}−29​ when xxx is 6 and yyy is 4 . What is xxx when yyy is 6 and zzz is −02-\frac{0}{2}−20​ ?

Studdy Solution

STEP 1

STEP 2

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

STEP 3

We're told that zzz varies directly as xxx and inversely as yyy.This translates to z=kxyz = \frac{kx}{y}z=ykx​, where kkk is our **constant of variation**.We need to find this kkk before we can do anything else!

STEP 4

We know that zzz is −92-\frac{9}{2}−29​ when xxx is 6 and yyy is 4.Let's **plug these values** into our equation: −92=k⋅64.-\frac{9}{2} = \frac{k \cdot 6}{4}.−29​=4k⋅6​.

STEP 5

To **isolate** kkk, we can multiply both sides of the equation by 4, which gives us: −92⋅4=k⋅6.-\frac{9}{2} \cdot 4 = k \cdot 6.−29​⋅4=k⋅6. −18=k⋅6.-18 = k \cdot 6.−18=k⋅6.

STEP 6

Now, **divide both sides by 6**: −186=k.\frac{-18}{6} = k.6−18​=k. k=−3.k = -3.k=−3.So our **constant of variation**, kkk, is −3-3−3!

STEP 7

Now we know our complete equation: z=−3xyz = \frac{-3x}{y}z=y−3x​.We want to find xxx when yyy is 6 and zzz is −02-\frac{0}{2}−20​.Notice that −02-\frac{0}{2}−20​ is just 0!Let's **plug in** these values: 0=−3x6.0 = \frac{-3x}{6}.0=6−3x​.

STEP 8

To **solve for** xxx, we can multiply both sides by 6: 0⋅6=−3x.0 \cdot 6 = -3x.0⋅6=−3x. 0=−3x.0 = -3x.0=−3x.

STEP 9

Finally, **divide both sides by** −3-3−3: 0−3=x.\frac{0}{-3} = x.−30​=x. x=0.x = 0.x=0.There it is!

STEP 10

When yyy is 6 and zzz is 0, xxx is 0.

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QuestionSay $z$ varies directly as $x$ and inversely as $y . z$ is $-\frac{9}{2}$ when $x$ is 6 and $y$ is 4 . What is $x$ when $y$ is 6 and $z$ is $-\frac{0}{2}$ ?

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

We're told that $z$ varies directly as $x$ and inversely as $y$ .
This translates to $z = \frac{kx}{y}$ , where $k$ is our constant of variation.
We need to find this $k$ before we can do anything else!

We know that $z$ is $-\frac{9}{2}$ when $x$ is 6 and $y$ is 4.
Let's plug these values into our equation: $-\frac{9}{2} = \frac{k \cdot 6}{4}.$

To isolate $k$ , we can multiply both sides of the equation by 4, which gives us: $-\frac{9}{2} \cdot 4 = k \cdot 6.$ $-18 = k \cdot 6.$

Now, divide both sides by 6: $\frac{-18}{6} = k.$ $k = -3.$ So our constant of variation, $k$ , is $-3$ !

Now we know our complete equation: $z = \frac{-3x}{y}$ .
We want to find $x$ when $y$ is 6 and $z$ is $-\frac{0}{2}$ .
Notice that $-\frac{0}{2}$ is just 0!
Let's plug in these values: $0 = \frac{-3x}{6}.$

To solve for $x$ , we can multiply both sides by 6: $0 \cdot 6 = -3x.$ $0 = -3x.$

Finally, divide both sides by $-3$ : $\frac{0}{-3} = x.$ $x = 0.$ There it is!

When $y$ is 6 and $z$ is 0, $x$ is 0.