Question

multiple choice 1 point a systems of equation is given: y = |x - 3| y = \frac{1}{2}x what is the distance between the points of intersection of the system? \sqrt{6} 4\sqrt{5} 4\sqrt{3} 2\sqrt{5}

Explanation:

Step1: Find intersection points

Set \(|x - 3|=\frac{1}{2}x\). We consider two cases:

• Case 1: \(x - 3=\frac{1}{2}x\) (when \(x\geq3\))

Subtract \(\frac{1}{2}x\) and add 3: \(\frac{1}{2}x = 3\), so \(x = 6\). Then \(y=\frac{1}{2}(6)=3\). Point: \((6, 3)\).

• Case 2: \(-(x - 3)=\frac{1}{2}x\) (when \(x<3\))

Simplify: \(-x + 3=\frac{1}{2}x\), add \(x\): \(3=\frac{3}{2}x\), so \(x = 2\). Then \(y=\frac{1}{2}(2)=1\). Point: \((2, 1)\).

Step2: Calculate distance

Use distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
Let \((x_1,y_1)=(2,1)\), \((x_2,y_2)=(6,3)\).
\(d=\sqrt{(6 - 2)^2+(3 - 1)^2}=\sqrt{4^2+2^2}=\sqrt{16 + 4}=\sqrt{20}=2\sqrt{5}\).

Answer:

\(2\sqrt{5}\) (corresponding to the option with \(2\sqrt{5}\))