Linear equations are fundamental in algebra, representing relationships with variables in standard, slope-intercept, and point-slope forms. They are used to graph straight lines, calculate slopes, and solve for variables. Understanding these equations is crucial for solving word problems and writing equations of parallel lines, making them a key concept in mathematics.
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Linear equations are algebraic expressions with variables raised to the first power and coefficients that represent the numbers multiplied by the variables
One-Variable Linear Equations
One-variable linear equations are typically written in the form "ax + b = 0", where 'a' is a non-zero coefficient
Two-Variable Linear Equations
Two-variable linear equations are typically written in the form "ax + by = c", where 'a' and 'b' are non-zero coefficients
Three-Variable Linear Equations
Three-variable linear equations are typically written in the form "ax + by + cz = d", where 'a', 'b', and 'c' are non-zero coefficients
The slope-intercept form is "y = mx + b", where 'm' is the slope and 'b' is the y-intercept, while the point-slope form is "y - y1 = m(x - x1)", where 'm' is the slope and '(x1, y1)' is a known point on the line
Solving linear equations involves finding the value(s) of the variable(s) that satisfy the equation
Linear equations can be constructed from two points by calculating the slope and using it to find the y-intercept
Word problems involving linear equations require converting narrative information into algebraic expressions and solving for the unknown quantities
Linear equations can be graphed on the Cartesian coordinate system, with one-variable equations resulting in straight lines
Parallel lines have the same slope, and their equations can be determined by converting the given line's equation into the slope-intercept form and finding the y-intercept
Linear equations can be expressed in function form, where 'y' is replaced with 'f(x)' to indicate that 'y' is a function of 'x'
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The solution to a linear equation involves finding the value(s) that make the equation ______.
satisfy
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Non-zero coefficients in linear equations
Coefficients 'a', 'b', and 'c' must be non-zero to maintain equation linearity and avoid trivial solutions.
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Purpose of standard form in linear equations
Standard form simplifies solving equations and lays groundwork for more complex algebra.
