5. Point \( P \) divides the directed line segment from point \( A(-4,-1) \) to point \( B(6,4) \) in the ratio \( 2: 3 \). What are the coordinates of point \( P \) ?

To find the coordinates of point \( P \) that divides the directed line segment from point \( A(-4,-1) \) to point \( B(6,4) \) in the ratio \( 2:3 \), we can use the section formula. The section formula states that if a point \( P(x, y) \) divides the line segment joining the points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \), then the coordinates of point \( P \) are given by: \[ x = \frac{mx_2 + nx_1}{m+n} \] \[ y = \frac{my_2 + ny_1}{m+n} \] Given that the ratio is \( 2:3 \), we can substitute the values of \( m = 2 \) and \( n = 3 \) into the section formula. Substitute the values of \( A(-4,-1) \) and \( B(6,4) \) into the section formula: \[ x = \frac{2(6) + 3(-4)}{2+3} \] \[ y = \frac{2(4) + 3(-1)}{2+3} \] Now, we can calculate the coordinates of point \( P \) using the section formula. Calculate the value by following steps: - step0: Calculate: \(\frac{\left(2\times 6+3\left(-4\right)\right)}{\left(2+3\right)}\) - step1: Remove the parentheses: \(\frac{2\times 6+3\left(-4\right)}{2+3}\) - step2: Multiply the numbers: \(\frac{12+3\left(-4\right)}{2+3}\) - step3: Multiply the numbers: \(\frac{12-12}{2+3}\) - step4: Subtract the numbers: \(\frac{0}{2+3}\) - step5: Add the numbers: \(\frac{0}{5}\) - step6: Divide the terms: \(0\) Calculate or simplify the expression \( (2*4 + 3*(-1))/(2+3) \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(2\times 4+3\left(-1\right)\right)}{\left(2+3\right)}\) - step1: Remove the parentheses: \(\frac{2\times 4+3\left(-1\right)}{2+3}\) - step2: Multiply the numbers: \(\frac{8+3\left(-1\right)}{2+3}\) - step3: Simplify: \(\frac{8-3}{2+3}\) - step4: Subtract the numbers: \(\frac{5}{2+3}\) - step5: Add the numbers: \(\frac{5}{5}\) - step6: Divide the terms: \(1\)