QuestionFind $DE$ if $DF = 42$ , $DE = 7x + 1$ , $EF = 4x - 3$ , and $DF = DE + EF$ .

Studdy Solution

STEP 1

Assumptions1. The length of line segment $DF$ is $42$ units. . The length of line segment $DE$ is $7x +1$ units.
3. The length of line segment $EF$ is $4x -3$ units.
4. The length of line segment $DF$ is equal to the sum of the lengths of line segments $DE$ and $EF$ .

STEP 2

First, we need to set up the equation based on the given assumption that $DF = DE + EF$ .
$DF = DE + EF$

STEP 3

Now, plug in the given values for $DF$ , $DE$ , and $EF$ into the equation.
$42 =7x +1 +x -3$

STEP 4

implify the equation by combining like terms.
$42 =11x -2$

STEP 5

To solve for $x$ , first isolate the term with $x$ by adding $2$ to both sides of the equation.
$42 +2 =11x -2 +2$

STEP 6

implify the equation.
$44 =11x$

STEP 7

Now, divide both sides of the equation by $11$ to solve for $x$ .
$x = \frac{44}{11}$

STEP 8

implify the equation to find the value of $x$ .
$x =4$

STEP 9

Now that we have the value of $x$ , we can substitute it into the expression for $DE$ to find the value of $DE$ .
$DE =7x +$

STEP 10

Substitute $x =4$ into the equation.
$DE =7(4) +$

STEP 11

implify the equation to find the value of $DE$ .
$DE =28 + =29$ The length of line segment $DE$ is $29$ units.

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QuestionFind $DE$ if $DF = 42$ , $DE = 7x + 1$ , $EF = 4x - 3$ , and $DF = DE + EF$ .

Studdy Solution

STEP 1

Assumptions1. The length of line segment $DF$ is $42$ units. . The length of line segment $DE$ is $7x +1$ units.
3. The length of line segment $EF$ is $4x -3$ units.
4. The length of line segment $DF$ is equal to the sum of the lengths of line segments $DE$ and $EF$ .

STEP 2

First, we need to set up the equation based on the given assumption that $DF = DE + EF$ .
$DF = DE + EF$

STEP 3

Now, plug in the given values for $DF$ , $DE$ , and $EF$ into the equation.
$42 =7x +1 +x -3$

STEP 4

implify the equation by combining like terms.
$42 =11x -2$

STEP 5

To solve for $x$ , first isolate the term with $x$ by adding $2$ to both sides of the equation.
$42 +2 =11x -2 +2$

STEP 6

implify the equation.
$44 =11x$

STEP 7

Now, divide both sides of the equation by $11$ to solve for $x$ .
$x = \frac{44}{11}$

STEP 8

implify the equation to find the value of $x$ .
$x =4$

STEP 9

Now that we have the value of $x$ , we can substitute it into the expression for $DE$ to find the value of $DE$ .
$DE =7x +$

STEP 10

Substitute $x =4$ into the equation.
$DE =7(4) +$

STEP 11

implify the equation to find the value of $DE$ .
$DE =28 + =29$ The length of line segment $DE$ is $29$ units.

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QuestionFind DEDEDE if DF=42DF = 42DF=42, DE=7x+1DE = 7x + 1DE=7x+1, EF=4x−3EF = 4x - 3EF=4x−3, and DF=DE+EFDF = DE + EFDF=DE+EF.

Studdy Solution

STEP 1

Assumptions1. The length of line segment DFDFDF is 424242 units. . The length of line segment DEDEDE is 7x+17x +17x+1 units.3. The length of line segment EFEFEF is 4x−34x -34x−3 units.4. The length of line segment DFDFDF is equal to the sum of the lengths of line segments DEDEDE and EFEFEF.

STEP 2

First, we need to set up the equation based on the given assumption that DF=DE+EFDF = DE + EFDF=DE+EF.DF=DE+EFDF = DE + EFDF=DE+EF

STEP 3

Now, plug in the given values for DFDFDF, DEDEDE, and EFEFEF into the equation.42=7x+1+x−342 =7x +1 +x -342=7x+1+x−3

STEP 4

implify the equation by combining like terms.42=11x−242 =11x -242=11x−2

STEP 5

To solve for xxx, first isolate the term with xxx by adding 222 to both sides of the equation.42+2=11x−2+242 +2 =11x -2 +242+2=11x−2+2

STEP 6

implify the equation.44=11x44 =11x44=11x

STEP 7

Now, divide both sides of the equation by 111111 to solve for xxx.x=4411x = \frac{44}{11}x=1144​

STEP 8

implify the equation to find the value of xxx.x=4x =4x=4

STEP 9

Now that we have the value of xxx, we can substitute it into the expression for DEDEDE to find the value of DEDEDE.DE=7x+DE =7x +DE=7x+

STEP 10

Substitute x=4x =4x=4 into the equation.DE=7(4)+DE =7(4) +DE=7(4)+

STEP 11

implify the equation to find the value of DEDEDE.DE=28+=29DE =28 + =29DE=28+=29The length of line segment DEDEDE is 292929 units.

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QuestionFind $DE$ if $DF = 42$ , $DE = 7x + 1$ , $EF = 4x - 3$ , and $DF = DE + EF$ .

Assumptions1. The length of line segment $DF$ is $42$ units. . The length of line segment $DE$ is $7x +1$ units.
3. The length of line segment $EF$ is $4x -3$ units.
4. The length of line segment $DF$ is equal to the sum of the lengths of line segments $DE$ and $EF$ .

First, we need to set up the equation based on the given assumption that $DF = DE + EF$ .
$DF = DE + EF$

Now, plug in the given values for $DF$ , $DE$ , and $EF$ into the equation.
$42 =7x +1 +x -3$

implify the equation by combining like terms.
$42 =11x -2$

To solve for $x$ , first isolate the term with $x$ by adding $2$ to both sides of the equation.
$42 +2 =11x -2 +2$

implify the equation.
$44 =11x$

Now, divide both sides of the equation by $11$ to solve for $x$ .
$x = \frac{44}{11}$

implify the equation to find the value of $x$ .
$x =4$

Now that we have the value of $x$ , we can substitute it into the expression for $DE$ to find the value of $DE$ .
$DE =7x +$

Substitute $x =4$ into the equation.
$DE =7(4) +$

implify the equation to find the value of $DE$ .
$DE =28 + =29$ The length of line segment $DE$ is $29$ units.