Answer:
Point-slope form: y + 1 = 2(x + 4)
Step-by-step explanation:
The given problem asks us to determine the point-slope form of the line passing through the points, (-4, -1) and (-3, 1).
Definitions:
The point-slope form of linear equations is: y - y₁ = m(x - x₁), where:
- m (slope) → it represents the steepness of a line. The slope also characterizes the ratio of the net change between the y- and x-coordinates. In other words, the slope of a line is the ratio between the vertical change in y-values ( rise ) over the horizontal change in x-values ( run ).
- (x₁ , y₁) → one of the points on the graph.
Solution:
Solve for the slope of the line:
First, we must solve for the slope of the line by using the following slope formula:
[tex]\displaystyle\mathsf{Slope\:(m)\:=\:\frac{y_2\:-\:y_1}{x_2\:-\:x_1}}}[/tex]
Let (x₁ , y₁) = (-4, -1)
(x₂, y₂) = (-3, 1)
Substitute these values into the slope formula:
[tex]\displaystyle\mathsf{Slope\:(m)\:=\:\frac{y_2\:-\:y_1}{x_2\:-\:x_1}}\:=\:\frac{\:1-(-1)}{-3-(-4)}\:=\:\frac{1+1}{-3+4}\:=\:\frac{2}{1}\:=\:2}[/tex]
Hence, the slope of the line is 2/1 or 2.
Write the point-slope form :
Substitute the slope and one of the given points into the point-slope form: y - y₁ = m(x - x₁):
⇒ y - y₁ = m(x - x₁)
⇒ y - (-1) = 2[x - (-4)]
Simplify:
⇒ y + 1 = 2(x + 4)
Final Answer:
Therefore, our point-slope form is: y + 1 = 2(x + 4).
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Additional notes:
Alternatively, we can use the other given point, (-3, 1). If we substitute this point into the point-slope form, then we will have the following equation:
slope (m) = 2
(x₁ , y₁) = (-3, 1)
y - ( 1 ) = 2[x - (-3)]
y - 1 = 2(x + 3) ⇒ This is the other point-slope form.
This equation has the same graph as the other point-slope form, y + 1 = 2(x + 4).
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Keywords:
Linear equations
Point-slope form
Slope
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