Question Find the values of $x$ and $y$ that satisfy the system of equations $x=7y$ and $y^{3}=x$ with $y \geq 0$ .

Studdy Solution

STEP 1

Assumptions
1. The relationship between $x$ and $y$ is given by $x=7y$ .
2. Another relationship is given by $y^3=x$ .
3. We are looking for solutions where $y \geq 0$ .

STEP 2

Since we have two equations involving $x$ and $y$ , we can use substitution to solve for one variable in terms of the other. We will substitute the expression for $x$ from the first equation into the second equation.
$y^3 = x$
$y^3 = 7y$

STEP 3

Now we solve the equation $y^3 = 7y$ . First, we can factor out a $y$ from both terms on the right-hand side.
$y^3 = y \cdot 7$
$y^3 = 7y$
$y^2 = 7$

STEP 4

Next, we solve for $y$ by taking the square root of both sides of the equation. Remember that we are only considering $y \geq 0$ .
$y = \sqrt{7}$

STEP 5

Now that we have the value of $y$ , we can substitute it back into the first equation to find the value of $x$ .
$x = 7y$
$x = 7\sqrt{7}$

STEP 6

We have found the values of $x$ and $y$ that satisfy both equations:
$x = 7\sqrt{7}$ $y = \sqrt{7}$
These values satisfy the conditions $x=7y$ and $y^3=x$ with $y \geq 0$ .

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Question Find the values of $x$ and $y$ that satisfy the system of equations $x=7y$ and $y^{3}=x$ with $y \geq 0$ .

Studdy Solution

STEP 1

Assumptions
1. The relationship between $x$ and $y$ is given by $x=7y$ .
2. Another relationship is given by $y^3=x$ .
3. We are looking for solutions where $y \geq 0$ .

STEP 2

Since we have two equations involving $x$ and $y$ , we can use substitution to solve for one variable in terms of the other. We will substitute the expression for $x$ from the first equation into the second equation.
$y^3 = x$
$y^3 = 7y$

STEP 3

Now we solve the equation $y^3 = 7y$ . First, we can factor out a $y$ from both terms on the right-hand side.
$y^3 = y \cdot 7$
$y^3 = 7y$
$y^2 = 7$

STEP 4

Next, we solve for $y$ by taking the square root of both sides of the equation. Remember that we are only considering $y \geq 0$ .
$y = \sqrt{7}$

STEP 5

Now that we have the value of $y$ , we can substitute it back into the first equation to find the value of $x$ .
$x = 7y$
$x = 7\sqrt{7}$

STEP 6

We have found the values of $x$ and $y$ that satisfy both equations:
$x = 7\sqrt{7}$ $y = \sqrt{7}$
These values satisfy the conditions $x=7y$ and $y^3=x$ with $y \geq 0$ .

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Question Find the values of xxx and yyy that satisfy the system of equations x=7yx=7yx=7y and y3=xy^{3}=xy3=x with y≥0y \geq 0y≥0.

Studdy Solution

STEP 1

Assumptions1. The relationship between xxx and yyy is given by x=7yx=7yx=7y.2. Another relationship is given by y3=xy^3=xy3=x.3. We are looking for solutions where y≥0y \geq 0y≥0.

STEP 2

Since we have two equations involving xxx and yyy, we can use substitution to solve for one variable in terms of the other. We will substitute the expression for xxx from the first equation into the second equation.y3=xy^3 = xy3=xy3=7yy^3 = 7yy3=7y

STEP 3

Now we solve the equation y3=7yy^3 = 7yy3=7y. First, we can factor out a yyy from both terms on the right-hand side.y3=y⋅7y^3 = y \cdot 7y3=y⋅7y3=7yy^3 = 7yy3=7yy2=7y^2 = 7y2=7

STEP 4

Next, we solve for yyy by taking the square root of both sides of the equation. Remember that we are only considering y≥0y \geq 0y≥0.y=7y = \sqrt{7}y=7​

STEP 5

Now that we have the value of yyy, we can substitute it back into the first equation to find the value of xxx.x=7yx = 7yx=7yx=77x = 7\sqrt{7}x=77​

STEP 6

We have found the values of xxx and yyy that satisfy both equations:x=77x = 7\sqrt{7}x=77​ y=7y = \sqrt{7}y=7​These values satisfy the conditions x=7yx=7yx=7y and y3=xy^3=xy3=x with y≥0y \geq 0y≥0.

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Question Find the values of $x$ and $y$ that satisfy the system of equations $x=7y$ and $y^{3}=x$ with $y \geq 0$ .

Assumptions
1. The relationship between $x$ and $y$ is given by $x=7y$ .
2. Another relationship is given by $y^3=x$ .
3. We are looking for solutions where $y \geq 0$ .

Since we have two equations involving $x$ and $y$ , we can use substitution to solve for one variable in terms of the other. We will substitute the expression for $x$ from the first equation into the second equation.
$y^3 = x$
$y^3 = 7y$

Now we solve the equation $y^3 = 7y$ . First, we can factor out a $y$ from both terms on the right-hand side.
$y^3 = y \cdot 7$
$y^3 = 7y$
$y^2 = 7$

Next, we solve for $y$ by taking the square root of both sides of the equation. Remember that we are only considering $y \geq 0$ .
$y = \sqrt{7}$

Now that we have the value of $y$ , we can substitute it back into the first equation to find the value of $x$ .
$x = 7y$
$x = 7\sqrt{7}$

We have found the values of $x$ and $y$ that satisfy both equations:
$x = 7\sqrt{7}$ $y = \sqrt{7}$
These values satisfy the conditions $x=7y$ and $y^3=x$ with $y \geq 0$ .