QuestionFind the length of $A C$ if line $y$ bisects $A C$ , $A B=4-5 x$ , and $B C=2 x+25$ .

Studdy Solution

STEP 1

Assumptions1. Line $y$ bisects $AC$ which means $AB = BC$ . . The length of $AB$ is given by $4-5x$ .
3. The length of $BC$ is given by $x+25$ .

STEP 2

Since $AB = BC$ , we can set the expressions for $AB$ and $BC$ equal to each other and solve for $x$ .
$4 -5x =2x +25$

STEP 3

To isolate $x$ , we can first add $5x$ to both sides of the equation.
$=2x +5x +25$

STEP 4

implify the right side of the equation.
$4 =7x +25$

STEP 5

Next, subtract $25$ from both sides of the equation to isolate $x$ .
$4 -25 =7x$

STEP 6

implify the left side of the equation.
$-21 =x$

STEP 7

Finally, divide both sides of the equation by $7$ to solve for $x$ .
$x = -21 /7$

STEP 8

Calculate the value of $x$ .
$x = -21 /7 = -3$

STEP 9

Now that we have the value of $x$ , we can find the length of $AC$ by adding the lengths of $AB$ and $BC$ . $AC = AB + BC$

STEP 10

Substitute the expressions for $AB$ and $BC$ into the equation.
$AC = (4 -5x) + (2x +25)$

STEP 11

Substitute $x = -3$ into the equation.
$AC = (4 -5(-3)) + ((-3) +25)$

STEP 12

implify the equation.
$AC = (4 +15) + (-6 +25)$

STEP 13

Calculate the length of $AC$ .
$AC =19 +19 =38$ The length of $AC$ is $38$ units.

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