Question Solve for $t$ in the equation $\frac{1}{r^{2}} = \frac{x+y}{t} + 8$ .

Studdy Solution

STEP 1

Assumptions
1. The equation given is $\frac{1}{r^{2}}=\frac{x+y}{t}+8$ .
2. We need to solve for $t$ .
3. All variables except $t$ are considered to be known quantities.

STEP 2

Isolate the term containing $t$ on one side of the equation by subtracting $8$ from both sides.
$\frac{1}{r^{2}} - 8 = \frac{x+y}{t}$

STEP 3

To solve for $t$ , we need to express the equation in terms of $\frac{1}{t}$ . To do this, we can take the reciprocal of both sides of the equation.
$\frac{1}{\frac{1}{r^{2}} - 8} = t \cdot \frac{1}{x+y}$

STEP 4

To simplify the left side of the equation, we can find a common denominator for the expression inside the reciprocal.
$\frac{1}{\frac{1}{r^{2}} - \frac{8r^{2}}{r^{2}}} = t \cdot \frac{1}{x+y}$

STEP 5

Combine the terms in the numerator on the left side of the equation.
$\frac{1}{\frac{1 - 8r^{2}}{r^{2}}} = t \cdot \frac{1}{x+y}$

STEP 6

Now, take the reciprocal of the fraction in the denominator on the left side to simplify further.
$\frac{r^{2}}{1 - 8r^{2}} = t \cdot \frac{1}{x+y}$

STEP 7

Multiply both sides of the equation by $\frac{1}{x+y}$ to isolate $t$ .
$\frac{r^{2}}{(1 - 8r^{2})(x+y)} = t$

STEP 8

Finally, we have the expression for $t$ in terms of $r$ , $x$ , $y$ .
$t = \frac{r^{2}}{(1 - 8r^{2})(x+y)}$
This is the solution for $t$ in the given equation.

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Question Solve for $t$ in the equation $\frac{1}{r^{2}} = \frac{x+y}{t} + 8$ .

Studdy Solution

STEP 1

Assumptions
1. The equation given is $\frac{1}{r^{2}}=\frac{x+y}{t}+8$ .
2. We need to solve for $t$ .
3. All variables except $t$ are considered to be known quantities.

STEP 2

Isolate the term containing $t$ on one side of the equation by subtracting $8$ from both sides.
$\frac{1}{r^{2}} - 8 = \frac{x+y}{t}$

STEP 3

To solve for $t$ , we need to express the equation in terms of $\frac{1}{t}$ . To do this, we can take the reciprocal of both sides of the equation.
$\frac{1}{\frac{1}{r^{2}} - 8} = t \cdot \frac{1}{x+y}$

STEP 4

To simplify the left side of the equation, we can find a common denominator for the expression inside the reciprocal.
$\frac{1}{\frac{1}{r^{2}} - \frac{8r^{2}}{r^{2}}} = t \cdot \frac{1}{x+y}$

STEP 5

Combine the terms in the numerator on the left side of the equation.
$\frac{1}{\frac{1 - 8r^{2}}{r^{2}}} = t \cdot \frac{1}{x+y}$

STEP 6

Now, take the reciprocal of the fraction in the denominator on the left side to simplify further.
$\frac{r^{2}}{1 - 8r^{2}} = t \cdot \frac{1}{x+y}$

STEP 7

Multiply both sides of the equation by $\frac{1}{x+y}$ to isolate $t$ .
$\frac{r^{2}}{(1 - 8r^{2})(x+y)} = t$

STEP 8

Finally, we have the expression for $t$ in terms of $r$ , $x$ , $y$ .
$t = \frac{r^{2}}{(1 - 8r^{2})(x+y)}$
This is the solution for $t$ in the given equation.

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Question Solve for ttt in the equation 1r2=x+yt+8\frac{1}{r^{2}} = \frac{x+y}{t} + 8r21​=tx+y​+8.

Studdy Solution

STEP 1

Assumptions1. The equation given is 1r2=x+yt+8\frac{1}{r^{2}}=\frac{x+y}{t}+8r21​=tx+y​+8.2. We need to solve for ttt.3. All variables except ttt are considered to be known quantities.

STEP 2

Isolate the term containing ttt on one side of the equation by subtracting 888 from both sides.1r2−8=x+yt\frac{1}{r^{2}} - 8 = \frac{x+y}{t}r21​−8=tx+y​

STEP 3

To solve for ttt, we need to express the equation in terms of 1t\frac{1}{t}t1​. To do this, we can take the reciprocal of both sides of the equation.11r2−8=t⋅1x+y\frac{1}{\frac{1}{r^{2}} - 8} = t \cdot \frac{1}{x+y}r21​−81​=t⋅x+y1​

STEP 4

To simplify the left side of the equation, we can find a common denominator for the expression inside the reciprocal.11r2−8r2r2=t⋅1x+y\frac{1}{\frac{1}{r^{2}} - \frac{8r^{2}}{r^{2}}} = t \cdot \frac{1}{x+y}r21​−r28r2​1​=t⋅x+y1​

STEP 5

Combine the terms in the numerator on the left side of the equation.11−8r2r2=t⋅1x+y\frac{1}{\frac{1 - 8r^{2}}{r^{2}}} = t \cdot \frac{1}{x+y}r21−8r2​1​=t⋅x+y1​

STEP 6

Now, take the reciprocal of the fraction in the denominator on the left side to simplify further.r21−8r2=t⋅1x+y\frac{r^{2}}{1 - 8r^{2}} = t \cdot \frac{1}{x+y}1−8r2r2​=t⋅x+y1​

STEP 7

Multiply both sides of the equation by 1x+y\frac{1}{x+y}x+y1​ to isolate ttt.r2(1−8r2)(x+y)=t\frac{r^{2}}{(1 - 8r^{2})(x+y)} = t(1−8r2)(x+y)r2​=t

STEP 8

Finally, we have the expression for ttt in terms of rrr, xxx, yyy.t=r2(1−8r2)(x+y)t = \frac{r^{2}}{(1 - 8r^{2})(x+y)}t=(1−8r2)(x+y)r2​This is the solution for ttt in the given equation.

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Question Solve for $t$ in the equation $\frac{1}{r^{2}} = \frac{x+y}{t} + 8$ .

Assumptions
1. The equation given is $\frac{1}{r^{2}}=\frac{x+y}{t}+8$ .
2. We need to solve for $t$ .
3. All variables except $t$ are considered to be known quantities.

Isolate the term containing $t$ on one side of the equation by subtracting $8$ from both sides.
$\frac{1}{r^{2}} - 8 = \frac{x+y}{t}$

To solve for $t$ , we need to express the equation in terms of $\frac{1}{t}$ . To do this, we can take the reciprocal of both sides of the equation.
$\frac{1}{\frac{1}{r^{2}} - 8} = t \cdot \frac{1}{x+y}$

To simplify the left side of the equation, we can find a common denominator for the expression inside the reciprocal.
$\frac{1}{\frac{1}{r^{2}} - \frac{8r^{2}}{r^{2}}} = t \cdot \frac{1}{x+y}$

Combine the terms in the numerator on the left side of the equation.
$\frac{1}{\frac{1 - 8r^{2}}{r^{2}}} = t \cdot \frac{1}{x+y}$

Now, take the reciprocal of the fraction in the denominator on the left side to simplify further.
$\frac{r^{2}}{1 - 8r^{2}} = t \cdot \frac{1}{x+y}$

Multiply both sides of the equation by $\frac{1}{x+y}$ to isolate $t$ .
$\frac{r^{2}}{(1 - 8r^{2})(x+y)} = t$

Finally, we have the expression for $t$ in terms of $r$ , $x$ , $y$ .
$t = \frac{r^{2}}{(1 - 8r^{2})(x+y)}$
This is the solution for $t$ in the given equation.