Solved on Jan 09, 2024

Find the value of $x-y$ given $xy=144$ , $x+y=30$ , and $x>y$ .

STEP 1

Assumptions
1. $x$ and $y$ are two numbers such that $xy = 144$ .
2. The sum of $x$ and $y$ is $x+y = 30$ .
3. $x$ is greater than $y$ ( $x > y$ ).
4. We are looking to find the value of $x-y$ .

STEP 2

We can use the system of equations to find the values of $x$ and $y$ . The equations are:
$\begin{align} xy &= 144, \\ x + y &= 30. \end{align}$

STEP 3

We can express $y$ in terms of $x$ using the second equation:
$y = 30 - x.$

STEP 4

Substitute the expression for $y$ into the first equation:
$x(30 - x) = 144.$

STEP 5

Expand the equation to form a quadratic equation:
$30x - x^2 = 144.$

STEP 6

Rearrange the equation to standard quadratic form:
$x^2 - 30x + 144 = 0.$

STEP 7

Factor the quadratic equation:
$(x - 24)(x - 6) = 0.$

STEP 8

Solve for the values of $x$ :
$x - 24 = 0 \quad \text{or} \quad x - 6 = 0.$

STEP 9

Find the two possible values of $x$ :
$x = 24 \quad \text{or} \quad x = 6.$

STEP 10

Since we know that $x > y$ , we take the larger value for $x$ :
$x = 24.$

STEP 11

Now, use the value of $x$ to find $y$ using the equation $x + y = 30$ :
$24 + y = 30.$

STEP 12

Solve for $y$ :
$y = 30 - 24.$

STEP 13

Calculate the value of $y$ :
$y = 6.$

STEP 14

Now that we have both $x$ and $y$ , we can find $x - y$ :
$x - y = 24 - 6.$

STEP 15

Calculate the value of $x - y$ :
$x - y = 18.$
The value of $x - y$ is 18.

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Find the value of $x-y$ given $xy=144$ , $x+y=30$ , and $x>y$ .

STEP 1

Assumptions
1. $x$ and $y$ are two numbers such that $xy = 144$ .
2. The sum of $x$ and $y$ is $x+y = 30$ .
3. $x$ is greater than $y$ ( $x > y$ ).
4. We are looking to find the value of $x-y$ .

STEP 2

We can use the system of equations to find the values of $x$ and $y$ . The equations are:
$\begin{align} xy &= 144, \\ x + y &= 30. \end{align}$

STEP 3

We can express $y$ in terms of $x$ using the second equation:
$y = 30 - x.$

STEP 4

Substitute the expression for $y$ into the first equation:
$x(30 - x) = 144.$

STEP 5

Expand the equation to form a quadratic equation:
$30x - x^2 = 144.$

STEP 6

Rearrange the equation to standard quadratic form:
$x^2 - 30x + 144 = 0.$

STEP 7

Factor the quadratic equation:
$(x - 24)(x - 6) = 0.$

STEP 8

Solve for the values of $x$ :
$x - 24 = 0 \quad \text{or} \quad x - 6 = 0.$

STEP 9

Find the two possible values of $x$ :
$x = 24 \quad \text{or} \quad x = 6.$

STEP 10

Since we know that $x > y$ , we take the larger value for $x$ :
$x = 24.$

STEP 11

Now, use the value of $x$ to find $y$ using the equation $x + y = 30$ :
$24 + y = 30.$

STEP 12

Solve for $y$ :
$y = 30 - 24.$

STEP 13

Calculate the value of $y$ :
$y = 6.$

STEP 14

Now that we have both $x$ and $y$ , we can find $x - y$ :
$x - y = 24 - 6.$

STEP 15

Calculate the value of $x - y$ :
$x - y = 18.$
The value of $x - y$ is 18.

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Find the value of x−yx-yx−y given xy=144xy=144xy=144, x+y=30x+y=30x+y=30, and x>yx>yx>y.

STEP 1

Assumptions1. xxx and yyy are two numbers such that xy=144xy = 144xy=144.2. The sum of xxx and yyy is x+y=30x+y = 30x+y=30.3. xxx is greater than yyy (x>yx > yx>y).4. We are looking to find the value of x−yx-yx−y.

STEP 2

We can use the system of equations to find the values of xxx and yyy. The equations are:xy=144,x+y=30. \begin{align*} xy &= 144, \\ x + y &= 30. \end{align*} xyx+y​=144,=30.​

STEP 3

We can express yyy in terms of xxx using the second equation:y=30−x. y = 30 - x. y=30−x.

STEP 4

Substitute the expression for yyy into the first equation:x(30−x)=144. x(30 - x) = 144. x(30−x)=144.

STEP 5

Expand the equation to form a quadratic equation:30x−x2=144. 30x - x^2 = 144. 30x−x2=144.

STEP 6

Rearrange the equation to standard quadratic form:x2−30x+144=0. x^2 - 30x + 144 = 0. x2−30x+144=0.

STEP 7

Factor the quadratic equation:(x−24)(x−6)=0. (x - 24)(x - 6) = 0. (x−24)(x−6)=0.

STEP 8

Solve for the values of xxx:x−24=0orx−6=0. x - 24 = 0 \quad \text{or} \quad x - 6 = 0. x−24=0orx−6=0.

STEP 9

Find the two possible values of xxx:x=24orx=6. x = 24 \quad \text{or} \quad x = 6. x=24orx=6.

STEP 10

Since we know that x>yx > yx>y, we take the larger value for xxx:x=24. x = 24. x=24.

STEP 11

Now, use the value of xxx to find yyy using the equation x+y=30x + y = 30x+y=30:24+y=30. 24 + y = 30. 24+y=30.

STEP 12

Solve for yyy:y=30−24. y = 30 - 24. y=30−24.

STEP 13

Calculate the value of yyy:y=6. y = 6. y=6.

STEP 14

Now that we have both xxx and yyy, we can find x−yx - yx−y:x−y=24−6. x - y = 24 - 6. x−y=24−6.

STEP 15

Calculate the value of x−yx - yx−y:x−y=18. x - y = 18. x−y=18.The value of x−yx - yx−y is 18.

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Find the value of $x-y$ given $xy=144$ , $x+y=30$ , and $x>y$ .

Assumptions
1. $x$ and $y$ are two numbers such that $xy = 144$ .
2. The sum of $x$ and $y$ is $x+y = 30$ .
3. $x$ is greater than $y$ ( $x > y$ ).
4. We are looking to find the value of $x-y$ .

We can use the system of equations to find the values of $x$ and $y$ . The equations are:
$\begin{align} xy &= 144, \\ x + y &= 30. \end{align}$

We can express $y$ in terms of $x$ using the second equation:
$y = 30 - x.$

Substitute the expression for $y$ into the first equation:
$x(30 - x) = 144.$

Expand the equation to form a quadratic equation:
$30x - x^2 = 144.$

Rearrange the equation to standard quadratic form:
$x^2 - 30x + 144 = 0.$

Factor the quadratic equation:
$(x - 24)(x - 6) = 0.$

Solve for the values of $x$ :
$x - 24 = 0 \quad \text{or} \quad x - 6 = 0.$

Find the two possible values of $x$ :
$x = 24 \quad \text{or} \quad x = 6.$

Since we know that $x > y$ , we take the larger value for $x$ :
$x = 24.$

Now, use the value of $x$ to find $y$ using the equation $x + y = 30$ :
$24 + y = 30.$

Solve for $y$ :
$y = 30 - 24.$

Calculate the value of $y$ :
$y = 6.$

Now that we have both $x$ and $y$ , we can find $x - y$ :
$x - y = 24 - 6.$

Calculate the value of $x - y$ :
$x - y = 18.$
The value of $x - y$ is 18.