The slope of a line between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
m = (y₂ - y₁) / (x₂ - x₁)
This formula calculates the rate of change of \(y\) with respect to \(x\), also known as "rise over run".
The equation of a line can be written as:
y = mx + b
Where:
If you know a point \((x₁, y₁)\) on the line and the slope \(m\), you can use the point-slope form:
y - y₁ = m(x - x₁)
This form is useful when you have a point and the slope, and you want to find the equation of the line.
The slope of a line can also be related to the angle of inclination \(\theta\). If you know the angle, the slope is:
m = tan(θ)
Where \(\theta\) is the angle the line makes with the x-axis.
If two lines are perpendicular, the product of their slopes is -1:
m₁ × m₂ = -1
Where m₁ and m₂ are the slopes of the two perpendicular lines.
If two lines are parallel, their slopes are equal:
m₁ = m₂
Where m₁ and m₂ are the slopes of the two parallel lines.
For horizontal and vertical lines:
If two lines are parallel, they have the same slope. For example:
y₁ = m₁x + b₁
y₂ = m₁x + b₂
Both lines have the same slope \(m₁\).
If you know the distance between two points and the x and y differences, you can find the slope using:
m = (Δy / Δx)
Where \(Δy\) is the difference in the y-coordinates and \(Δx\) is the difference in the x-coordinates.
The general form of the equation of a line is:
Ax + By + C = 0
Where A, B, and C are constants. You can rearrange this to find the slope.
Solution: Using the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \):
\( m = \frac{8 - 4}{7 - 3} = \frac{4}{4} = 1 \)
The slope is \( m = 1 \).
Solution: Slope \( m = \tan(\theta) \):
\( m = \tan(30^\circ) \approx 0.577 \)
The slope is approximately \( m = 0.577 \).
Solution: First, calculate the slope of the original line:
\( m = \frac{15 - 5}{8 - 2} = \frac{10}{6} = \frac{5}{3} \)
The slope of any line parallel to this is also \( m = \frac{5}{3} \).
Solution: The slope of a perpendicular line is the negative reciprocal of the original slope:
\( m_{\text{perpendicular}} = -\frac{5}{4} \)
The perpendicular slope is \( m = -\frac{5}{4} \).
Solution: Rewrite in slope-intercept form \( y = mx + b \):
\( 3x - 4y = 12 \rightarrow y = \frac{3}{4}x - 3 \)
The slope is \( m = \frac{3}{4} \).
Solution: Slope \( m = \frac{\text{rise}}{\text{run}} \):
\( m = \frac{10}{3} \)
The slope is \( m = \frac{10}{3} \).
Solution: This is a vertical line, so the slope is undefined.
Since \( x_1 = x_2 \), the slope \( m = \frac{y_2 - y_1}{x_2 - x_1} \) results in division by zero.
Solution: The angle \( \theta = \arctan(m) \):
\( \theta = \arctan(-1.5) \approx -56.31^\circ \)
Solution: The slope of the original line is \( m = -2 \).
The slope of a perpendicular line is \( m = \frac{1}{2} \).
Solution: Calculate the slope:
\( m = \frac{8 - 2}{4 - 1} = 2 \)
Since \( m > 0 \), the line is increasing.
Solution: Using the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \):
\( m = \frac{5 - (-3)}{2 - (-4)} = \frac{8}{6} = \frac{4}{3} \)
The slope is \( m = \frac{4}{3} \).
Solution: The slope of the given line is \( m = 5 \). Any parallel line will have the same slope.
The slope of the parallel line is \( m = 5 \).
Solution: The slope of the given line is \( m = 3 \). The slope of a perpendicular line is the negative reciprocal:
\( m_{\text{perpendicular}} = -\frac{1}{3} \)
The perpendicular slope is \( m = -\frac{1}{3} \).
Solution: Rewrite the equation in slope-intercept form \( y = mx + b \):
\( 2x + 3y = 6 \rightarrow y = -\frac{2}{3}x + 2 \)
The slope is \( m = -\frac{2}{3} \).
Solution: Using the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \( (x_1, y_1) = (0, 0) \):
\( m = \frac{8 - 0}{6 - 0} = \frac{8}{6} = \frac{4}{3} \)
The slope is \( m = \frac{4}{3} \).
Solution: The equation is already in slope-intercept form \( y = mx + b \), so the slope is \( m = -4 \).
The slope is \( m = -4 \).
Solution: Using the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \):
\( m = \frac{6 - (-2)}{-3 - 5} = \frac{8}{-8} = -1 \)
The slope is \( m = -1 \).
Solution: Using the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \):
\( m = \frac{-2 - 3}{4 - (-1)} = \frac{-5}{5} = -1 \)
The slope is \( m = -1 \).
Solution: Using the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \):
\( m = \frac{10 - 0}{10 - 0} = \frac{10}{10} = 1 \)
The slope is \( m = 1 \).
Solution: Using the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \):
\( m = \frac{-2 - (-4)}{6 - 8} = \frac{2}{-2} = -1 \)
The slope is \( m = -1 \).
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