Question

practice problems

1. what is an equation for the line that passes through the coordinates (2,7) and (9,1)?

1. what is an equation for the line that passes through the coordinates (2,8) and (0,3)?

1. what is an equation for the line that passes through the coordinates (4,2) and (7,6)?

1. find the equation of the line that passes through the points (1,1) and (3,5)?

1. find the equation of the line that passes through the points (1,3) and (2,4)?

Explanation:

Let's solve problem 4: Find the equation of the line that passes through the points \((1,1)\) and \((3,5)\).

Step 1: Calculate the slope (\(m\))

The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Let \((x_1, y_1) = (1, 1)\) and \((x_2, y_2) = (3, 5)\).
\[
m = \frac{5 - 1}{3 - 1} = \frac{4}{2} = 2
\]

Step 2: Use point - slope form to find the equation

The point - slope form of a line is \(y - y_1 = m(x - x_1)\). We can use the point \((1,1)\) and \(m = 2\).
\[
y - 1 = 2(x - 1)
\]

Step 3: Simplify to slope - intercept form (\(y=mx + b\))

Expand the right - hand side:
\[
y - 1 = 2x-2
\]
Add 1 to both sides:
\[
y=2x - 2 + 1
\]
\[
y = 2x-1
\]

Answer:

The equation of the line is \(y = 2x - 1\) (or in standard form \(2x-y=1\))