Question

complete the point-slope equation of the line through (-9, 6) and (-7, -8). use exact numbers. y - 6 =

Explanation:

Step1: Find the slope (m)

The slope formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Here, \((x_1, y_1)=(-9, 6)\) and \((x_2, y_2)=(-7, -8)\). So, \(m=\frac{-8 - 6}{-7 - (-9)}=\frac{-14}{2}=-7\).

Step2: Use point - slope form

The point - slope form of a line is \(y - y_1=m(x - x_1)\). We know \(y_1 = 6\), \(x_1=-9\) and \(m=-7\). Substituting these values into the point - slope form, we get \(y - 6=-7(x - (-9))=-7(x + 9)\). In the given equation \(y - 6=\square\), the value in the square is \(-7(x + 9)\) (but if we just consider the slope - part for the box as per the equation structure, since the point - slope is \(y - y_1=m(x - x_1)\), and here \(y_1 = 6\), \(x_1=-9\), the slope \(m=-7\) and the \((x - x_1)\) part is \((x+9)\), so the expression in the box is \(-7(x + 9)\), but if we are to write the slope - multiplied by \((x - x_1)\), and the question has \(y - 6=\square\), the \(\square\) should be \(-7(x + 9)\), but if we consider the slope first, the slope is \(-7\) and then \((x - (-9))=(x + 9)\), so the complete expression is \(-7(x + 9)\). However, if we look at the equation \(y - 6=\square\), and the point - slope formula is \(y - y_1=m(x - x_1)\), so substituting \(m=-7\) and \(x_1=-9\), we have \(y - 6=-7(x - (-9))=-7(x + 9)\). But if we just need the coefficient and the \((x - x_1)\) part, the slope \(m=-7\) and \(x_1=-9\), so the expression is \(-7(x + 9)\). But maybe the question is asking for the slope - multiplied by \((x - x_1)\), so the value in the box is \(-7(x + 9)\). But if we simplify the slope calculation, we found \(m=-7\), and the point - slope form with the given point \((-9,6)\) is \(y - 6=-7(x + 9)\). So the expression in the box is \(-7(x + 9)\), but if we consider the equation as \(y - 6=m(x - x_1)\), and we need to fill the box, the box should have \(-7(x + 9)\). But if we are to write the slope - part, the slope is \(-7\) and the \((x - x_1)\) is \((x+9)\), so the answer for the box is \(-7(x + 9)\). But if we just look at the numerical coefficient and the \((x - x_1)\) structure, the slope \(m=-7\) and \(x_1=-9\), so the expression is \(-7(x + 9)\).

Answer:

\(-7(x + 9)\) (or if we consider the slope - multiplied by \((x - x_1)\) and the equation \(y - 6=\square\), the value in the box is \(-7(x + 9)\))