Question

ex. 3 if ac = 40, ab = 6x - 3, and bc = 2x - 5, find x, ab and bc
ex. 4 if ab = 1000, ab = 12x - 500, and bc = 8x - 300, find x, ab and bc
ex. 5 - draw: in a line segment point y is between x and z, xy = 2x + 1, yz = 6x and xz = 81; find the value of x and yz.

Explanation:

Ex. 3

Step1: Set up equation based on segment addition

Since $AC = AB+BC$, we have $40=(6x - 3)+(2x - 5)$.

Step2: Simplify the right - hand side

Combine like terms: $40 = 6x+2x-3 - 5$, so $40=8x - 8$.

Step3: Solve for x

Add 8 to both sides: $40 + 8=8x$, i.e., $48 = 8x$. Then divide both sides by 8, $x=\frac{48}{8}=6$.

Step4: Find AB

Substitute $x = 6$ into the formula for $AB$: $AB=6x - 3=6\times6 - 3=36 - 3 = 33$.

Step5: Find BC

Substitute $x = 6$ into the formula for $BC$: $BC=2x - 5=2\times6 - 5=12 - 5 = 7$.

Step1: Solve for x using the given AB value

Since $AB = 12x-500$ and $AB = 1000$, we set up the equation $12x-500 = 1000$.

Step2: Isolate the term with x

Add 500 to both sides: $12x=1000 + 500=1500$.

Step3: Solve for x

Divide both sides by 12: $x=\frac{1500}{12}=125$.

Step4: Find BC

Substitute $x = 125$ into the formula for $BC$: $BC=8x - 300=8\times125 - 300=1000 - 300 = 700$.

Step1: Set up equation based on segment addition

Since $XZ=XY + YZ$, we have $81=(2x + 1)+6x$.

Step2: Simplify the right - hand side

Combine like terms: $81=2x+6x + 1$, so $81=8x + 1$.

Step3: Solve for x

Subtract 1 from both sides: $81-1=8x$, i.e., $80 = 8x$. Then divide both sides by 8, $x = 10$.

Step4: Find YZ

Substitute $x = 10$ into the formula for $YZ$: $YZ=6x=6\times10 = 60$.

Answer:

$x = 6$, $AB = 33$, $BC = 7$

Ex. 4