Write an equation for each line in point-slope form and then convert it to standard form. Show your work.

Question

through (2, 3) and (3, 5)

Answer 1

To find the equation of a line given two points, we start by using the formula for point-slope form:

Point-slope form: y - y₁ = m(x - x₁)

Where (x₁, y₁) represents one of the points on the line, and m is the slope of the line.

First, let's find the slope (m):

m = (y₂ - y₁) / (x₂ - x₁)

Using the points (2, 3) and (3, 5):

m = (5 - 3) / (3 - 2)
m = 2 / 1
m = 2

Now we can substitute one of the points and the slope into the point-slope form equation. Let's use (2, 3):

y - 3 = 2(x - 2)

Expand and simplify:

y - 3 = 2x - 4
y = 2x - 1

So, the equation of the line in point-slope form is y = 2x - 1.

To convert the equation to standard form (Ax + By = C), we need to eliminate any fractions and rearrange the equation.

y = 2x - 1

First, let's move the 2x term to the other side:

-2x + y = -1

Next, let's multiply through by -1 to eliminate the negative sign:

2x - y = 1

Finally, let's multiply through by 1 to remove any fractions:

2x - y = 1

The equation of the line in standard form is 2x - y = 1.