Question

1. write an equation for a line parallel to the line defined by (9,4) and (16, 25) in point - slope form.

Explanation:

Step1: Find the slope of the given line

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \). For the points \((9, 4)\) and \((16, 25)\), we have \( x_1 = 9\), \( y_1 = 4\), \( x_2 = 16\), \( y_2 = 25 \). So, \( m=\frac{25 - 4}{16 - 9}=\frac{21}{7} = 3 \).

Step2: Recall the property of parallel lines

Parallel lines have the same slope. So, the slope of the line we want to find is also \( 3 \).

Step3: Use the point - slope form formula

The point - slope form of a line is \( y - y_1=m(x - x_1) \), where \((x_1, y_1)\) is a point on the line and \( m \) is the slope. We can use the point \((9, 4)\) (we could also use any other point, but we'll use this one for simplicity). Substituting \( m = 3\), \( x_1=9\) and \( y_1 = 4 \) into the point - slope form, we get \( y - 4=3(x - 9) \). (We could also use another point, for example, if we use a general point \((x_0,y_0)\), the equation would be \( y - y_0=3(x - x_0) \), but using \((9,4)\) is a valid choice here.)

Answer:

\( y - 4 = 3(x - 9) \) (or any equivalent point - slope form using a different point with slope 3, e.g., \( y - 25=3(x - 16) \))