Question

complete the point-slope equation of the line through (1, -1) and (5, 2). use exact numbers. y - (-1) =

Explanation:

Step1: Find the slope

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Here, \((x_1, y_1)=(1, - 1)\) and \((x_2, y_2)=(5, 2)\). So \( m=\frac{2-(-1)}{5 - 1}=\frac{2 + 1}{4}=\frac{3}{4} \).

Step2: Point - slope form

The point - slope form of a line is \( y - y_1=m(x - x_1) \). We use the point \((1,-1)\), so \( y_1=-1\) and \( x_1 = 1\), and \( m=\frac{3}{4} \). Substituting these values into the point - slope form, we get \( y-(-1)=\frac{3}{4}(x - 1) \). The part we need to fill in the box is the slope - point part, which is \(\frac{3}{4}(x - 1)\)? Wait, no, the left side is \( y-(-1)\), and the right side should be \( m(x - x_1) \). Since \( m=\frac{3}{4}\) and \( x_1 = 1\), the right side is \(\frac{3}{4}(x - 1)\)? Wait, no, the question is to complete \( y-(-1)=\square\), so the square should be \(\frac{3}{4}(x - 1)\)? Wait, no, let's re - check. The point - slope formula is \( y - y_1=m(x - x_1) \). We have \( y_1=-1\), so \( y-(-1)=m(x - 1) \), and \( m=\frac{3}{4}\), so \( y + 1=\frac{3}{4}(x - 1) \). So the box should be \(\frac{3}{4}(x - 1)\)? Wait, no, the problem shows \( y-(-1)=\square\), so the square is \(\frac{3}{4}(x - 1)\)? Wait, no, the slope is \(\frac{3}{4}\), and the point - slope form with the point \((1,-1)\) is \( y-(-1)=\frac{3}{4}(x - 1) \). So the expression to put in the box is \(\frac{3}{4}(x - 1)\)? Wait, no, maybe I made a mistake. Wait, the point - slope form is \( y - y_1=m(x - x_1) \). So if we take the point \((1,-1)\), then \( y-(-1)=m(x - 1) \), and \( m=\frac{3}{4}\), so the right - hand side is \(\frac{3}{4}(x - 1)\). But the problem is asking to complete \( y-(-1)=\square\), so the square should be \(\frac{3}{4}(x - 1)\)? Wait, no, maybe the question is to write the slope times \((x - x_1)\). Since \( m=\frac{3}{4}\) and \( x_1 = 1\), it's \(\frac{3}{4}(x - 1)\). But let's check the slope calculation again. \( 2-(-1)=3\), \( 5 - 1 = 4\), so slope is \(\frac{3}{4}\). Then the point - slope form using \((1,-1)\) is \( y+1=\frac{3}{4}(x - 1)\), so the part after the equal sign is \(\frac{3}{4}(x - 1)\).

Answer:

\(\frac{3}{4}(x - 1)\)