Linear equations are essential in Algebra and Trigonometry, representing relationships between variables. Understanding different forms, like slope-intercept and standard form, helps in graphing, solving systems, and applying these concepts to real-world problems effectively.
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Slope-intercept form (y = mx + b)
- The equation represents a straight line where 'm' is the slope and 'b' is the y-intercept.
- Slope (m) indicates the steepness and direction of the line (positive slope = rising, negative slope = falling).
- The y-intercept (b) is the point where the line crosses the y-axis (x=0).
- This form is useful for quickly graphing linear equations and understanding their behavior.
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Point-slope form
- The equation is written as y - y₁ = m(x - x₁), where (x₁, y₁) is a specific point on the line.
- It is particularly useful when you know a point on the line and the slope.
- This form allows for easy conversion to slope-intercept form for graphing.
- It emphasizes the relationship between a point on the line and the slope.
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Standard form (Ax + By = C)
- The equation is expressed as Ax + By = C, where A, B, and C are integers, and A should be non-negative.
- This form is useful for identifying intercepts and for solving systems of equations.
- It can represent vertical and horizontal lines, unlike slope-intercept form.
- Converting to this form can simplify the process of solving linear equations.
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Vertical and horizontal lines
- Vertical lines have the equation x = a (undefined slope) and run up and down.
- Horizontal lines have the equation y = b (zero slope) and run left to right.
- Understanding these lines is crucial for graphing and identifying relationships between variables.
- They serve as special cases in linear equations and can affect the solutions of systems.
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Parallel and perpendicular lines
- Parallel lines have the same slope (m) and never intersect.
- Perpendicular lines have slopes that are negative reciprocals of each other (m₁ * m₂ = -1).
- Recognizing these relationships helps in graphing and solving systems of equations.
- These concepts are essential in geometry and real-world applications.
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Systems of linear equations
- A system consists of two or more linear equations that share common variables.
- Solutions can be found using graphing, substitution, or elimination methods.
- The solution represents the point(s) where the lines intersect (if any).
- Systems can have one solution, no solution, or infinitely many solutions.
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Graphing linear equations
- To graph, identify the slope and y-intercept from the equation.
- Plot the y-intercept on the y-axis and use the slope to find another point.
- Draw a straight line through the points to represent the equation.
- Accurate graphing is essential for visualizing relationships and solutions.
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Finding x and y intercepts
- The x-intercept is found by setting y = 0 and solving for x.
- The y-intercept is found by setting x = 0 and solving for y.
- These intercepts are key points for graphing linear equations.
- They provide insight into the behavior of the equation and its solutions.
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Solving linear equations algebraically
- Use inverse operations to isolate the variable on one side of the equation.
- Combine like terms and simplify to find the solution.
- Check your solution by substituting back into the original equation.
- Mastery of algebraic techniques is crucial for solving more complex equations.
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Applications of linear equations in real-world problems
- Linear equations model relationships in various fields such as economics, physics, and biology.
- They can represent trends, forecasts, and optimization problems.
- Understanding how to apply linear equations helps in making informed decisions based on data.
- Real-world applications often involve interpreting the slope and intercept in context.