Solved on Dec 14, 2023

Find $x$ and $y$ given $xy=144$ , $x+y=30$ , and $x \geq y$ . What is the value of $x-y$ ?

STEP 1

Assumptions
1. The product of $x$ and $y$ is 144, i.e., $xy = 144$ .
2. The sum of $x$ and $y$ is 30, i.e., $x + y = 30$ .
3. $x$ is greater than or equal to $y$ , i.e., $x \geq y$ .
4. We need to find the value of $x - y$ .

STEP 2

We have a system of two equations with two variables:
\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 $y$ from STEP_3 into the first equation:
$x(30 - x) = 144$

STEP 5

Expand the equation and move all terms to one side to form a quadratic equation:
$x^2 - 30x + 144 = 0$

STEP 6

Factor the quadratic equation:
$(x - 12)(x - 18) = 0$

STEP 7

Set each factor equal to zero and solve for $x$ :
$x - 12 = 0 \quad \text{or} \quad x - 18 = 0$

STEP 8

Solve for $x$ in each equation:
$x = 12 \quad \text{or} \quad x = 18$

STEP 9

Since $x \geq y$ , and both $x$ and $y$ are positive, we choose the larger value for $x$ :
$x = 18$

STEP 10

Now, substitute $x = 18$ into the second equation to find $y$ :
$18 + y = 30$

STEP 11

Solve for $y$ :
$y = 30 - 18$

STEP 12

Calculate the value of $y$ :
$y = 12$

STEP 13

Now that we have both $x$ and $y$ , we can find the value of $x - y$ :
$x - y = 18 - 12$

STEP 14

Calculate the difference:
$x - y = 6$
The value of $x - y$ is 6.

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Find $x$ and $y$ given $xy=144$ , $x+y=30$ , and $x \geq y$ . What is the value of $x-y$ ?

STEP 1

Assumptions
1. The product of $x$ and $y$ is 144, i.e., $xy = 144$ .
2. The sum of $x$ and $y$ is 30, i.e., $x + y = 30$ .
3. $x$ is greater than or equal to $y$ , i.e., $x \geq y$ .
4. We need to find the value of $x - y$ .

STEP 2

We have a system of two equations with two variables:
\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 $y$ from STEP_3 into the first equation:
$x(30 - x) = 144$

STEP 5

Expand the equation and move all terms to one side to form a quadratic equation:
$x^2 - 30x + 144 = 0$

STEP 6

Factor the quadratic equation:
$(x - 12)(x - 18) = 0$

STEP 7

Set each factor equal to zero and solve for $x$ :
$x - 12 = 0 \quad \text{or} \quad x - 18 = 0$

STEP 8

Solve for $x$ in each equation:
$x = 12 \quad \text{or} \quad x = 18$

STEP 9

Since $x \geq y$ , and both $x$ and $y$ are positive, we choose the larger value for $x$ :
$x = 18$

STEP 10

Now, substitute $x = 18$ into the second equation to find $y$ :
$18 + y = 30$

STEP 11

Solve for $y$ :
$y = 30 - 18$

STEP 12

Calculate the value of $y$ :
$y = 12$

STEP 13

Now that we have both $x$ and $y$ , we can find the value of $x - y$ :
$x - y = 18 - 12$

STEP 14

Calculate the difference:
$x - y = 6$
The value of $x - y$ is 6.

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Find xxx and yyy given xy=144xy=144xy=144, x+y=30x+y=30x+y=30, and x≥yx \geq yx≥y. What is the value of x−yx-yx−y?

STEP 1

Assumptions1. The product of xxx and yyy is 144, i.e., xy=144xy = 144xy=144.2. The sum of xxx and yyy is 30, i.e., x+y=30x + y = 30x+y=30.3. xxx is greater than or equal to yyy, i.e., x≥yx \geq yx≥y.4. We need to find the value of x−yx - yx−y.

STEP 2

We have a system of two equations with two variables:\begin{align*} xy &= 144 \\ x + y &= 30 \end{align*}

STEP 3

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

STEP 4

Substitute the expression for yyy from STEP_3 into the first equation:x(30−x)=144x(30 - x) = 144x(30−x)=144

STEP 5

Expand the equation and move all terms to one side to form a quadratic equation:x2−30x+144=0x^2 - 30x + 144 = 0x2−30x+144=0

STEP 6

Factor the quadratic equation:(x−12)(x−18)=0(x - 12)(x - 18) = 0(x−12)(x−18)=0

STEP 7

Set each factor equal to zero and solve for xxx:x−12=0orx−18=0x - 12 = 0 \quad \text{or} \quad x - 18 = 0x−12=0orx−18=0

STEP 8

Solve for xxx in each equation:x=12orx=18x = 12 \quad \text{or} \quad x = 18x=12orx=18

STEP 9

Since x≥yx \geq yx≥y, and both xxx and yyy are positive, we choose the larger value for xxx:x=18x = 18x=18

STEP 10

Now, substitute x=18x = 18x=18 into the second equation to find yyy:18+y=3018 + y = 3018+y=30

STEP 11

Solve for yyy:y=30−18y = 30 - 18y=30−18

STEP 12

Calculate the value of yyy:y=12y = 12y=12

STEP 13

Now that we have both xxx and yyy, we can find the value of x−yx - yx−y:x−y=18−12x - y = 18 - 12x−y=18−12

STEP 14

Calculate the difference:x−y=6x - y = 6x−y=6The value of x−yx - yx−y is 6.

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Find $x$ and $y$ given $xy=144$ , $x+y=30$ , and $x \geq y$ . What is the value of $x-y$ ?

Assumptions
1. The product of $x$ and $y$ is 144, i.e., $xy = 144$ .
2. The sum of $x$ and $y$ is 30, i.e., $x + y = 30$ .
3. $x$ is greater than or equal to $y$ , i.e., $x \geq y$ .
4. We need to find the value of $x - y$ .

We have a system of two equations with two variables:
\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 $y$ from STEP_3 into the first equation:
$x(30 - x) = 144$

Expand the equation and move all terms to one side to form a quadratic equation:
$x^2 - 30x + 144 = 0$

Factor the quadratic equation:
$(x - 12)(x - 18) = 0$

Set each factor equal to zero and solve for $x$ :
$x - 12 = 0 \quad \text{or} \quad x - 18 = 0$

Solve for $x$ in each equation:
$x = 12 \quad \text{or} \quad x = 18$

Since $x \geq y$ , and both $x$ and $y$ are positive, we choose the larger value for $x$ :
$x = 18$

Now, substitute $x = 18$ into the second equation to find $y$ :
$18 + y = 30$

Solve for $y$ :
$y = 30 - 18$

Calculate the value of $y$ :
$y = 12$

Now that we have both $x$ and $y$ , we can find the value of $x - y$ :
$x - y = 18 - 12$

Calculate the difference:
$x - y = 6$
The value of $x - y$ is 6.