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Questions and Answers
What is the formula for calculating the slope m of a line passing through the points $(x_1, y_1)$ and $(x_2, y_2)$?
What is the formula for calculating the slope m of a line passing through the points $(x_1, y_1)$ and $(x_2, y_2)$?
What is the point-slope form of the equation of a line?
What is the point-slope form of the equation of a line?
If a line has a slope of $\frac{2}{3}$, what is the slope of a line parallel to it?
If a line has a slope of $\frac{2}{3}$, what is the slope of a line parallel to it?
In finding the intersection point of the equations $3x + 4y = 8$ and $6x - 10y = 7$, what variable was solved for first?
In finding the intersection point of the equations $3x + 4y = 8$ and $6x - 10y = 7$, what variable was solved for first?
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What kind of line will have a slope that is the negative reciprocal of the slope of the line $3x + 4y = 8$?
What kind of line will have a slope that is the negative reciprocal of the slope of the line $3x + 4y = 8$?
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When the first equation $3x + 4y = 8$ was multiplied by $-2$ and added to the second equation, what was the purpose of this operation?
When the first equation $3x + 4y = 8$ was multiplied by $-2$ and added to the second equation, what was the purpose of this operation?
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Given a slope of 3, which of the following points would result in the equation passing through point $(1, -2)$?
Given a slope of 3, which of the following points would result in the equation passing through point $(1, -2)$?
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To write the equation of a line that is perpendicular to the line $3x + 4y = 8$, which slope would you need to use?
To write the equation of a line that is perpendicular to the line $3x + 4y = 8$, which slope would you need to use?
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Study Notes
Lines in a Plane
- The slope of a non-vertical line passing through points (x₁, y₁) and (x₂, y₂) is calculated as: m = (y₂ - y₁) / (x₂ - x₁)
- x₁ ≠ x₂
- The order of subtraction is crucial in calculating the slope.
- A positive slope indicates a line rising from left to right.
- A zero slope indicates a horizontal line.
- A negative slope indicates a line falling from left to right.
- An undefined slope indicates a vertical line.
Slope of Lines through Specific Points
- Example:
- Find the slope of the line passing through (-2, 0) and (3, 1).
- m = (1 - 0) / (3 - (-2)) = 1/5
- Find the slope of the line passing through (-1, 2) and (2, 2).
- m = (2 - 2) / (2 - (-1)) = 0/3 = 0
- Find the slope of the line passing through (0, 4) and (1, -1).
- m = (-1 - 4) / (1 - 0) = -5/1 = -5
- Find the slope of the line passing through (3, 4) and (3, 1).
- m = (1 - 4) / (3 - 3) = -3/0 (undefined)
Possibilities for a Line's Slope
- Positive slope: the line rises from left to right.
- Negative slope: the line falls from left to right.
- Zero slope: the line is horizontal.
- Undefined slope: the line is vertical.
The Point-Slope Form
- The equation of a line passing through (x₁, y₁) with slope m is: y - y₁ = m(x - x₁)
Example of Point-Slope Form
- Find an equation of the line that passes through (1, -2) and has a slope of 3.
- Using the point-slope form: y - (-2) = 3(x - 1).
- Simplifying: y + 2 = 3x - 3.
- The equation: y = 3x - 5
Slope-Intercept Form
- The equation of a line in slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept.
Summary of Equations of Lines
- General form: Ax + By + C = 0
- Vertical line: x = a
- Horizontal line: y = b
- Slope-intercept form: y = mx + b
- Point-slope form: y - y₁ = m(x - x₁)
Example of Finding Slope and y-intercept
- Find the slope and y-intercept of -4y = 5x - 6.
- Rewrite in slope-intercept form: y = (-5/4)x + (3/2).
- Slope: -5/4, y-intercept: 3/2
Parallel and Perpendicular Lines
- Parallel lines have equal slopes.
- Perpendicular lines have slopes that are negative reciprocals of each other.
Example of Parallel Lines
- Find an equation of the line passing through (2, -1) and parallel to 2x - 3y = 5.
- Rewrite 2x - 3y = 5 in slope-intercept form to find the slope of the parallel line.
- Slope is 2/3.
- Use the point-slope form with (x₁, y₁) = (2, -1) and m = 2/3: y - (-1) = (2/3)(x - 2) y + 1 = (2/3)x - 4/3
Example of finding an equation of a perpendicular line
- Find an equation of the line passing through (2, -1) which is perpendicular to 2x - 3y = 5.
- Rewrite 2x − 3y = 5 in slope intercept form.
- The slope is 2/3.
- The perpendicular line's slope is −3/2.
- Using the point-slope form with (x₁,y₁) = (2, −1) and m = -3/2 y − (-1) = (-3/2)(x − 2) y = (-3/2)x + 2
Finding the Intersection of Two Lines
- To find the intersection of two lines (e.g., 3x + 4y = 8 and 6x − 10y = 7), solve the system of equations.
Value of 'c' for Specific Line Properties
- Find the value of c for which the line 3x + cy = 5 satisfies the given conditions (e.g., passes through a point, parallel to a line, etc.). This involves substituting the given values and solving for c.
- For example, find c if it passes through (3, 1).
Finding 'k' in an equation
- Find k in kx − 3y = 10 if it meets specific conditions (e.g., parallel to, perpendicular to another line).
- This involves finding the slope from the equations and using the conditions to solve for k.
Determining if a point lies above or below a line
- Given a point and a line equation, substitute the coordinates of the point into the equation to determine if the point's y-value is greater than or less than the line's y-value at that x.
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Description
Test your understanding of lines in a plane and calculate slopes with this quiz. You'll explore positive, negative, zero, and undefined slopes through various examples, enhancing your comprehension of line behavior on a graph.
