Answer:
Slope-intercept form: y = x - 11
Standard form: y - x = 11
Point-slope form: y + 5 = (x - 6)
Step-by-step explanation:
Given the two points, (5, -6) and (6, -5):
We must first solve for the slope, as it will be used in the point-slope and slope-intercept forms.
Slope
Let (x₁, y₁) = (5, -6)
(x₂, y₂) = (6, -5)
Substitute these values into the following slope formula:
[tex]\LARGE\mathsf{m = \frac {(y_2\: -\: y_1)}{(x_2\: -\: x_1)}}[/tex]
[tex]\LARGE\mathsf{m = \frac {-5\: -\: (-6)}{6\:-\:5}\:=\frac{-5\:+\:6}{ 1 } \:=\frac{1}{ 1 }}[/tex]
m = 1
Therefore, the slope of the line is 1.
Y-intercept
Next, we must determine the y-intercept of the line. The y-intercept is the point on the graph where it crosses the y-axis, at point (0, b). In order to find the y-intercept, use one of the given points, (6, -5), and the slope, m = 1, and substitute into the slope-intercept form, y = mx + b:
y = mx + b
-5 = 1 (6) + b
-5 = 6 + b
Subtract 6 from both sides:
-5 - 6 = 6 - 6 + b
-11 = b
Therefore, the y-intercept is (0, -11), where b = -11.
Now that we have the value for the slope of the line, m = 1, and the y-intercept, b = -11, we could proceed in creating the linear equation in slope-intercept, point-slope, and standard forms.
Given our slope, m = 1, and the y-intercept, b = -11, we can express the linear equation in the following forms:
Slope-intercept form: y = x - 11
Standard Form:
We could transform the slope-intercept form into the standard form:
Ax + By = C
where:
- A is a non-negative integer.
Transform the slope-intercept form into standard form.
Start by subtracting x from both sides of the equation:
y - x = x - x - 11
y - x = -11 ⇒ This is the standard form where A = 1, B = -1, and C = -11.
Point-slope form:
Use one of the given points, (6, -5) and substitute into the point-slope form:
y - y₁ = m(x - x₁)
y - (-5) = 1(x - 6)
y + 5 = (x - 6) ⇒ This is the point-slope form.
