Linear equations are mathematical statements that describe a relationship between two variables in a way that produces a straight line when graphed. The general form of a linear equation in two variables (x and y) can be written as:
\[ y = mx + b \]
Where:
- \( y \) is the dependent variable.
- \( x \) is the independent variable.
- \( m \) is the slope of the line (the rate of change) that describes how much \( y \) changes for a one-unit change in \( x \).
- \( b \) is the y-intercept, the value of \( y \) when \( x = 0 \).
Key Concepts:
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Slope (m): Determines the steepness and direction of the line:
- Positive slope (\( m > 0 \)): The line rises as it moves from left to right.
- Negative slope (\( m < 0 \)): The line falls as it moves from left to right.
- Zero slope (\( m = 0 \)): The line is horizontal.
- Undefined slope: The line is vertical.
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Y-intercept (b): The point at which the line crosses the y-axis.
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Standard Form: A linear equation can also be expressed in standard form: \[ Ax + By = C \] where \( A \), \( B \), and \( C \) are constants, and \( A \) and \( B \) are not both zero.
Examples:
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Example 1:
- Equation: \( y = 2x + 3 \)
- Slope \( m = 2 \), y-intercept \( b = 3 \)
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Example 2:
- Equation: \( 4x - 2y = 8 \)
- To express in slope-intercept form, solve for \( y \):
- \( -2y = -4x + 8 \)
- \( y = 2x - 4 \) (Here, \( m = 2 \) and \( b = -4 \))
Solving Linear Equations:
To solve linear equations, you can use various methods, including substitution, elimination, and graphing.
Example of Solving via Substitution:
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Given a system of equations:
- \( y = 2x + 3 \)
- \( x + y = 9 \)
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Substitute the expression for \( y \) from Equation 1 into Equation 2: \[ x + (2x + 3) = 9 \] \[ 3x + 3 = 9 \] \[ 3x = 6 \quad \Rightarrow \quad x = 2 \]
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Now substitute \( x \) back into Equation 1: \[ y = 2(2) + 3 = 7 \]
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Solution: \( x = 2, y = 7 \)
If you have more specific questions or examples you'd like to explore regarding linear equations, feel free to ask!
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