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Questions and Answers
What is the general form of a linear equation in two variables, and what do the constants A, B, and C represent?
What is the general form of a linear equation in two variables, and what do the constants A, B, and C represent?
The general form of a linear equation in two variables is Ax + By = C, where A, B, and C are constants, and A and B are not both zero. The constants A and B represent the coefficients of the variables x and y, respectively, and C represents the constant term.
How is the slope of a line calculated, and what does a positive slope indicate?
How is the slope of a line calculated, and what does a positive slope indicate?
The slope of a line is calculated as m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line. A positive slope indicates a line that rises from left to right.
What is the purpose of the y-intercept in graphing a linear equation, and how is it found?
What is the purpose of the y-intercept in graphing a linear equation, and how is it found?
The y-intercept is the point where the line crosses the y-axis, and it is used as a starting point for graphing the equation. The y-intercept is found by plugging in x = 0 into the equation.
What is the difference between slope-intercept form and point-slope form of the equation of a line?
What is the difference between slope-intercept form and point-slope form of the equation of a line?
What does a zero slope indicate about a line, and what is the graph of a linear equation in two variables?
What does a zero slope indicate about a line, and what is the graph of a linear equation in two variables?
What is the purpose of finding the x-intercept in graphing a linear equation, and how is it found?
What is the purpose of finding the x-intercept in graphing a linear equation, and how is it found?
Match the following forms of linear equations with their characteristics:
Match the following forms of linear equations with their characteristics:
Match the following types of slopes with their descriptions:
Match the following types of slopes with their descriptions:
Match the following graphing methods with their formulas:
Match the following graphing methods with their formulas:
Match the following characteristics with the types of lines:
Match the following characteristics with the types of lines:
Match the following steps with the process of graphing a linear equation:
Match the following steps with the process of graphing a linear equation:
Match the following characteristics with the forms of linear equations:
Match the following characteristics with the forms of linear equations:
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Study Notes
Linear Equations in Two Variables
- A linear equation in two variables is an equation that can be written in the form:
- Ax + By = C
- Where A, B, and C are constants, and A and B are not both zero
- The graph of a linear equation in two variables is a straight line
- Solutions to the equation are the points (x, y) that satisfy the equation
Slope of a Line
- The slope of a line is a measure of how steep it is
- Slope is denoted by the letter m and is calculated as:
- m = (y2 - y1) / (x2 - x1)
- Where (x1, y1) and (x2, y2) are two points on the line
- Slope can be positive, negative, zero, or undefined
- A positive slope indicates a line that rises from left to right
- A negative slope indicates a line that falls from left to right
- A zero slope indicates a horizontal line
- An undefined slope indicates a vertical line
Graphing Linear Equations
- To graph a linear equation, start by plotting the y-intercept (the point where the line crosses the y-axis)
- Use the slope to find additional points on the line
- Plot the additional points and draw a straight line through them
- The x-intercept (the point where the line crosses the x-axis) can be found by plugging in x = 0 into the equation
Forms of the Equation of a Line
- There are three main forms of the equation of a line:
- Slope-Intercept Form: y = mx + b
- Where m is the slope and b is the y-intercept
- Point-Slope Form: y - y1 = m(x - x1)
- Where (x1, y1) is a point on the line and m is the slope
- Standard Form: Ax + By = C
- Where A, B, and C are constants, and A and B are not both zero
- Slope-Intercept Form: y = mx + b
Slopes of Parallel and Perpendicular Lines
- Parallel Lines: lines that never intersect and have the same slope
- If two lines are parallel, their slopes are equal
- Perpendicular Lines: lines that intersect at a right angle (90 degrees) and have slopes that are negative reciprocals
- If two lines are perpendicular, their slopes are negative reciprocals of each other
- If the slope of one line is m, the slope of a parallel line is also m, and the slope of a perpendicular line is -1/m
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