QuestionFind $x$ and the length of $PQ$ given $PQ=2x+1$ and $QR=5x-44$ , with $Q$ as the midpoint of $\overline{PR}$ .

Studdy Solution

STEP 1

Assumptions1. $Q$ is the midpoint of $\overline{PR}$ . . The length of segment $Q$ is given by the expression $x+1$ .
3. The length of segment $QR$ is given by the expression $5x-44$ .

STEP 2

Since $Q$ is the midpoint of $\overline{PR}$ , the lengths of segments $Q$ and $QR$ are equal. We can set up the equation $2x+1 =5x-44$ to solve for $x$ .

STEP 3

Subtract $2x$ from both sides of the equation to isolate the variable $x$ on one side.
$2x+1 =5x-44 \Rightarrow1 =3x -44$

STEP 4

Add $44$ to both sides of the equation to isolate $3x$ .
$1 =3x -44 \Rightarrow45 =3x$

STEP 5

Divide both sides of the equation by $3$ to solve for $x$ .
$45 =3x \Rightarrow x =15$

STEP 6

Now that we have the value of $x$ , we can substitute it into the expression for $Q$ to find the length of $Q$ .
$Q =2x+1$

STEP 7

Substitute $x =15$ into the equation to calculate the length of $Q$ .
$Q =2(15) +1$

STEP 8

Calculate the length of $Q$ .
$Q =2(15) +1 =31$ The value of $x$ is $15$ and the length of $Q$ is $31$ .

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QuestionFind $x$ and the length of $PQ$ given $PQ=2x+1$ and $QR=5x-44$ , with $Q$ as the midpoint of $\overline{PR}$ .

Studdy Solution

STEP 1

Assumptions1. $Q$ is the midpoint of $\overline{PR}$ . . The length of segment $Q$ is given by the expression $x+1$ .
3. The length of segment $QR$ is given by the expression $5x-44$ .

STEP 2

Since $Q$ is the midpoint of $\overline{PR}$ , the lengths of segments $Q$ and $QR$ are equal. We can set up the equation $2x+1 =5x-44$ to solve for $x$ .

STEP 3

Subtract $2x$ from both sides of the equation to isolate the variable $x$ on one side.
$2x+1 =5x-44 \Rightarrow1 =3x -44$

STEP 4

Add $44$ to both sides of the equation to isolate $3x$ .
$1 =3x -44 \Rightarrow45 =3x$

STEP 5

Divide both sides of the equation by $3$ to solve for $x$ .
$45 =3x \Rightarrow x =15$

STEP 6

Now that we have the value of $x$ , we can substitute it into the expression for $Q$ to find the length of $Q$ .
$Q =2x+1$

STEP 7

Substitute $x =15$ into the equation to calculate the length of $Q$ .
$Q =2(15) +1$

STEP 8

Calculate the length of $Q$ .
$Q =2(15) +1 =31$ The value of $x$ is $15$ and the length of $Q$ is $31$ .

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QuestionFind xxx and the length of PQPQPQ given PQ=2x+1PQ=2x+1PQ=2x+1 and QR=5x−44QR=5x-44QR=5x−44, with QQQ as the midpoint of PR‾\overline{PR}PR.

Studdy Solution

STEP 1

Assumptions1. QQQ is the midpoint of PR‾\overline{PR}PR. . The length of segment QQQ is given by the expression x+1x+1x+1.3. The length of segment QRQRQR is given by the expression 5x−445x-445x−44.

STEP 2

Since QQQ is the midpoint of PR‾\overline{PR}PR, the lengths of segments QQQ and QRQRQR are equal. We can set up the equation 2x+1=5x−442x+1 =5x-442x+1=5x−44 to solve for xxx.

STEP 3

Subtract 2x2x2x from both sides of the equation to isolate the variable xxx on one side.2x+1=5x−44⇒1=3x−442x+1 =5x-44 \Rightarrow1 =3x -442x+1=5x−44⇒1=3x−44

STEP 4

Add 444444 to both sides of the equation to isolate 3x3x3x.1=3x−44⇒45=3x1 =3x -44 \Rightarrow45 =3x1=3x−44⇒45=3x

STEP 5

Divide both sides of the equation by 333 to solve for xxx.45=3x⇒x=1545 =3x \Rightarrow x =1545=3x⇒x=15

STEP 6

Now that we have the value of xxx, we can substitute it into the expression for QQQ to find the length of QQQ.Q=2x+1Q =2x+1Q=2x+1

STEP 7

Substitute x=15x =15x=15 into the equation to calculate the length of QQQ.Q=2(15)+1Q =2(15) +1Q=2(15)+1

STEP 8

Calculate the length of QQQ.Q=2(15)+1=31Q =2(15) +1 =31Q=2(15)+1=31The value of xxx is 151515 and the length of QQQ is 313131.

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QuestionFind $x$ and the length of $PQ$ given $PQ=2x+1$ and $QR=5x-44$ , with $Q$ as the midpoint of $\overline{PR}$ .

Assumptions1. $Q$ is the midpoint of $\overline{PR}$ . . The length of segment $Q$ is given by the expression $x+1$ .
3. The length of segment $QR$ is given by the expression $5x-44$ .

Since $Q$ is the midpoint of $\overline{PR}$ , the lengths of segments $Q$ and $QR$ are equal. We can set up the equation $2x+1 =5x-44$ to solve for $x$ .

Subtract $2x$ from both sides of the equation to isolate the variable $x$ on one side.
$2x+1 =5x-44 \Rightarrow1 =3x -44$

Add $44$ to both sides of the equation to isolate $3x$ .
$1 =3x -44 \Rightarrow45 =3x$

Divide both sides of the equation by $3$ to solve for $x$ .
$45 =3x \Rightarrow x =15$

Now that we have the value of $x$ , we can substitute it into the expression for $Q$ to find the length of $Q$ .
$Q =2x+1$

Substitute $x =15$ into the equation to calculate the length of $Q$ .
$Q =2(15) +1$

Calculate the length of $Q$ .
$Q =2(15) +1 =31$ The value of $x$ is $15$ and the length of $Q$ is $31$ .