Write an equation for each line in point-slope form and then convert it to standard form. Show your work.
through (2, 3) and (3, 5)
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.