Question

points a, b, and c are collinear. ab = 2x+1, bc = 15, and ac = 5x-5. solve for x points a, b, and c are collinear. ab = 4x-1, bc = 5x+11, and ac = 100. solve for ab 20 solve for x

Explanation:

First Sub - Question (Solve for \(x\) when \(AB = 2x + 1\), \(BC=15\), \(AC = 5x-5\) and \(A\), \(B\), \(C\) are collinear)

Step 1: Use the collinearity property

Since \(A\), \(B\), \(C\) are collinear, \(AB + BC=AC\). Substitute the given expressions: \(2x + 1+15 = 5x-5\)

Step 2: Simplify the equation

Simplify the left - hand side: \(2x+16 = 5x - 5\)

Step 3: Move the \(x\) terms to one side

Subtract \(2x\) from both sides: \(16=5x - 2x-5\), which simplifies to \(16 = 3x-5\)

Step 4: Solve for \(x\)

Add 5 to both sides: \(16 + 5=3x\), so \(21 = 3x\). Divide both sides by 3: \(x=\frac{21}{3}=7\)

Second Sub - Question (Solve for \(AB\) when \(AB = 4x - 1\), \(BC = 5x+11\), \(AC = 100\) and \(A\), \(B\), \(C\) are collinear)

Step 1: Use the collinearity property

Since \(A\), \(B\), \(C\) are collinear, \(AB + BC=AC\). Substitute the given expressions: \(4x-1+5x + 11=100\)

Step 2: Simplify the equation

Combine like terms: \(9x + 10=100\)

Step 3: Solve for \(x\)

Subtract 10 from both sides: \(9x=100 - 10=90\). Divide both sides by 9: \(x = 10\)

Step 4: Find the value of \(AB\)

Substitute \(x = 10\) into the expression for \(AB\): \(AB=4\times10-1=40 - 1 = 39\)

Answer:

For the first sub - question (Solve for \(x\)): \(x = 7\)

For the second sub - question (Solve for \(AB\)): \(AB=39\)