Question

on a number line, the directed line segment from q to s has endpoints q at -2 and s at 6. point r partitions the directed line segment from q to s in a 3:2 ratio. rachel uses the section formula to find the location of point r on the number line. her work is shown below. let m = 3, n = 2, x₁ = -2, and x₂ = 6. 1. r = (m x₂ + n x₁)/(m + n) 2. r = (3(6) + 2(-2))/(3 + 2) what is the location of point r on the number line? options: 14/5, 16/5, 18/5, 22/5

Explanation:

Step1: Calculate numerator

Multiply \(3\) by \(6\) and \(2\) by \(-2\), then add.
\(3(6)+2(-2)=18 - 4 = 14\)

Step2: Calculate denominator

Add \(3\) and \(2\).
\(3 + 2 = 5\)

Step3: Find \(R\)

Divide numerator by denominator.
\(R=\frac{14}{5}\)? Wait, no, wait—wait, let's re - check. Wait, \(3\times6 = 18\), \(2\times(-2)=-4\), \(18+( - 4)=14\)? Wait, no, the formula is \(R=\frac{mx_2+nx_1}{m + n}\), \(m = 3\), \(x_2 = 6\), \(n = 2\), \(x_1=-2\). So \(mx_2=3\times6 = 18\), \(nx_1=2\times(-2)=-4\), sum is \(18+( - 4)=14\)? Wait, but the options have \(\frac{16}{5}\)? Wait, no, maybe I made a mistake. Wait, no, the ratio is from Q to S, so the section formula for internal division is \(\frac{mx_2+nx_1}{m + n}\) when dividing in the ratio \(m:n\) from the first point \(x_1\) to \(x_2\). Wait, Q is at \(-2\) (\(x_1\)), S is at \(6\) (\(x_2\)), ratio \(3:2\) (m:n = 3:2). So \(mx_2+nx_1=3\times6+2\times(-2)=18 - 4 = 14\), \(m + n=5\), so \(R=\frac{14}{5}\)? But wait, let's recalculate: \(3\times6 = 18\), \(2\times(-2)=-4\), \(18-4 = 14\), \(14\div5=\frac{14}{5}\). Wait, but the first option is \(\frac{14}{5}\). Wait, maybe I misread the ratio. Wait, the directed line segment from Q to S, so the ratio is m:n where m is the part near Q? No, the section formula: if a point divides the line segment joining \((x_1,y_1)\) and \((x_2,y_2)\) in the ratio \(m:n\) (from \((x_1,y_1)\) to \((x_2,y_2)\)), then the coordinates are \(\frac{mx_2+nx_1}{m + n},\frac{my_2+ny_1}{m + n}\). So in this case, on the number line, \(y_1=y_2 = 0\). So \(x\) - coordinate is \(\frac{3\times6+2\times(-2)}{3 + 2}=\frac{18-4}{5}=\frac{14}{5}\). Wait, but let's check again. \(3\times6 = 18\), \(2\times(-2)=-4\), \(18+( - 4)=14\), \(14\div5=\frac{14}{5}\). So the answer should be \(\frac{14}{5}\)? Wait, but the first option is \(\frac{14}{5}\). Wait, maybe I made a mistake earlier. Let's do the calculation step by step:

1. Calculate the numerator: \(3\times6+2\times(-2)=18 - 4 = 14\)
2. Calculate the denominator: \(3 + 2 = 5\)
3. Then \(R=\frac{14}{5}\)

Answer:

\(\frac{14}{5}\) (corresponding to the first option: \(\frac{14}{5}\))