Solved on Feb 17, 2024

Find the length of CD when given that CD = x + 8, DE = 3x + 1, and CE = 5x - 8. Express the answer as a simplified fraction, mixed number, or integer.

STEP 1

Assumptions
1. $CD = x + 8$
2. $DE = 3x + 1$
3. $CE = 5x - 8$
4. $CE = CD + DE$ because in a straight line, the length from $C$ to $E$ is the sum of the lengths from $C$ to $D$ and from $D$ to $E$ .

STEP 2

We will use the relationship between the segments to set up an equation.
$CE = CD + DE$

STEP 3

Substitute the expressions for $CD$ , $DE$ , and $CE$ into the equation.
$(5x - 8) = (x + 8) + (3x + 1)$

STEP 4

Combine like terms on the right side of the equation.
$(5x - 8) = x + 3x + 8 + 1$

STEP 5

Further simplify the right side by combining the $x$ terms and the constant terms.
$(5x - 8) = 4x + 9$

STEP 6

Subtract $4x$ from both sides of the equation to get the $x$ terms on one side.
$5x - 4x - 8 = 4x - 4x + 9$

STEP 7

Simplify both sides of the equation.
$x - 8 = 9$

STEP 8

Add $8$ to both sides of the equation to solve for $x$ .
$x - 8 + 8 = 9 + 8$

STEP 9

Simplify both sides of the equation.
$x = 17$

STEP 10

Now that we have the value of $x$ , we can find the length of segment $CD$ by substituting $x$ back into the expression for $CD$ .
$CD = x + 8$

STEP 11

Substitute $x = 17$ into the expression for $CD$ .
$CD = 17 + 8$

STEP 12

Calculate the value of $CD$ .
$CD = 25$
So, the length of segment $CD$ is $25$ .

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Find the length of CD when given that CD = x + 8, DE = 3x + 1, and CE = 5x - 8. Express the answer as a simplified fraction, mixed number, or integer.

STEP 1

Assumptions
1. $CD = x + 8$
2. $DE = 3x + 1$
3. $CE = 5x - 8$
4. $CE = CD + DE$ because in a straight line, the length from $C$ to $E$ is the sum of the lengths from $C$ to $D$ and from $D$ to $E$ .

STEP 2

We will use the relationship between the segments to set up an equation.
$CE = CD + DE$

STEP 3

Substitute the expressions for $CD$ , $DE$ , and $CE$ into the equation.
$(5x - 8) = (x + 8) + (3x + 1)$

STEP 4

Combine like terms on the right side of the equation.
$(5x - 8) = x + 3x + 8 + 1$

STEP 5

Further simplify the right side by combining the $x$ terms and the constant terms.
$(5x - 8) = 4x + 9$

STEP 6

Subtract $4x$ from both sides of the equation to get the $x$ terms on one side.
$5x - 4x - 8 = 4x - 4x + 9$

STEP 7

Simplify both sides of the equation.
$x - 8 = 9$

STEP 8

Add $8$ to both sides of the equation to solve for $x$ .
$x - 8 + 8 = 9 + 8$

STEP 9

Simplify both sides of the equation.
$x = 17$

STEP 10

Now that we have the value of $x$ , we can find the length of segment $CD$ by substituting $x$ back into the expression for $CD$ .
$CD = x + 8$

STEP 11

Substitute $x = 17$ into the expression for $CD$ .
$CD = 17 + 8$

STEP 12

Calculate the value of $CD$ .
$CD = 25$
So, the length of segment $CD$ is $25$ .

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Find the length of CD when given that CD = x + 8, DE = 3x + 1, and CE = 5x - 8. Express the answer as a simplified fraction, mixed number, or integer.

STEP 1

Assumptions1. CD=x+8CD = x + 8CD=x+82. DE=3x+1DE = 3x + 1DE=3x+13. CE=5x−8CE = 5x - 8CE=5x−84. CE=CD+DECE = CD + DECE=CD+DE because in a straight line, the length from CCC to EEE is the sum of the lengths from CCC to DDD and from DDD to EEE.

STEP 2

We will use the relationship between the segments to set up an equation.CE=CD+DECE = CD + DECE=CD+DE

STEP 3

Substitute the expressions for CDCDCD, DEDEDE, and CECECE into the equation.(5x−8)=(x+8)+(3x+1)(5x - 8) = (x + 8) + (3x + 1)(5x−8)=(x+8)+(3x+1)

STEP 4

Combine like terms on the right side of the equation.(5x−8)=x+3x+8+1(5x - 8) = x + 3x + 8 + 1(5x−8)=x+3x+8+1

STEP 5

Further simplify the right side by combining the xxx terms and the constant terms.(5x−8)=4x+9(5x - 8) = 4x + 9(5x−8)=4x+9

STEP 6

Subtract 4x4x4x from both sides of the equation to get the xxx terms on one side.5x−4x−8=4x−4x+95x - 4x - 8 = 4x - 4x + 95x−4x−8=4x−4x+9

STEP 7

Simplify both sides of the equation.x−8=9x - 8 = 9x−8=9

STEP 8

Add 888 to both sides of the equation to solve for xxx.x−8+8=9+8x - 8 + 8 = 9 + 8x−8+8=9+8

STEP 9

Simplify both sides of the equation.x=17x = 17x=17

STEP 10

Now that we have the value of xxx, we can find the length of segment CDCDCD by substituting xxx back into the expression for CDCDCD.CD=x+8CD = x + 8CD=x+8

STEP 11

Substitute x=17x = 17x=17 into the expression for CDCDCD.CD=17+8CD = 17 + 8CD=17+8

STEP 12

Calculate the value of CDCDCD.CD=25CD = 25CD=25So, the length of segment CDCDCD is 252525.

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Assumptions
1. $CD = x + 8$
2. $DE = 3x + 1$
3. $CE = 5x - 8$
4. $CE = CD + DE$ because in a straight line, the length from $C$ to $E$ is the sum of the lengths from $C$ to $D$ and from $D$ to $E$ .

We will use the relationship between the segments to set up an equation.
$CE = CD + DE$

Substitute the expressions for $CD$ , $DE$ , and $CE$ into the equation.
$(5x - 8) = (x + 8) + (3x + 1)$

Combine like terms on the right side of the equation.
$(5x - 8) = x + 3x + 8 + 1$

Further simplify the right side by combining the $x$ terms and the constant terms.
$(5x - 8) = 4x + 9$

Subtract $4x$ from both sides of the equation to get the $x$ terms on one side.
$5x - 4x - 8 = 4x - 4x + 9$

Simplify both sides of the equation.
$x - 8 = 9$

Add $8$ to both sides of the equation to solve for $x$ .
$x - 8 + 8 = 9 + 8$

Simplify both sides of the equation.
$x = 17$

Now that we have the value of $x$ , we can find the length of segment $CD$ by substituting $x$ back into the expression for $CD$ .
$CD = x + 8$

Substitute $x = 17$ into the expression for $CD$ .
$CD = 17 + 8$

Calculate the value of $CD$ .
$CD = 25$
So, the length of segment $CD$ is $25$ .