Examples of linear regression in the following topics:
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- Linear regression is an approach to modeling the linear relationship between a dependent variable, $y$ and an independent variable, $x$.
- With linear regression, a line in slope-intercept form, $y=mx+b$ is found that "best fits" the data.
- The simplest and perhaps most common linear regression model is the ordinary least squares approximation.
- Indeed, trying to fit linear models to data that is quadratic, cubic, or anything non-linear, or data with many outliers or errors can result in bad approximations.
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- A zero, or $x$-intercept, is the point at which a linear function's value will equal zero.
- The graph of a linear function is a straight line.
- Linear functions can have none, one, or infinitely many zeros.
- To find the zero of a linear function, simply find the point where the line crosses the $x$-axis.
- To find the zero of a linear function algebraically, set $y=0$ and solve for $x$.
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- Linear equations are those with one or more variables of the first order.
- There is in fact a field of mathematics known as linear algebra, in which linear equations in up to an infinite number of variables are studied.
- Linear equations can therefore be expressed in general (standard) form as:
- For example,imagine these linear equations represent the trajectories of two vehicles.
- Imagine these linear equations represent the trajectories of two vehicles.
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- A system of linear equations
consists of two or more linear equations made up of two or more
variables, such that all equations in the system are considered
simultaneously.
- Some linear systems may not have a solution, while others may have an
infinite number of solutions.
- For example, consider the following
system of linear equations in two variables:
- In this example, the ordered pair (4, 7) is the
solution to the system of linear equations.
- In general, a linear system may behave in any one of three possible ways:
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- A linear inequality is an expression that is designated as less than, greater than, less than or equal to, or greater than or equal to.
- When two linear expressions are not equal, but are designated as less than ($<$), greater than ($>$), less than or equal to ($\leq$) or greater than or equal to ($\geq$), it is called a linear inequality.
- A linear inequality looks exactly like a linear equation, with the inequality sign replacing the equality sign.
- A linear inequality looks like a linear equation, with the inequality sign replacing the equal sign.
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- Linear functions are algebraic equations whose graphs are straight lines with unique values for their slope and y-intercepts.
- A linear function is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable.
- The origin of the name "linear" comes from the fact that the set of solutions of such an equation forms a straight line in the plane.
- The blue line, $y=\frac{1}{2}x-3$ and the red line, $y=-x+5$ are both linear functions.
- Identify what makes a function linear and the characteristics of a linear function
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- Two properties of a linear system are consistency (are there solutions?
- In mathematics, a system of linear equations (or linear system) is a collection of linear equations involving the same set of variables.
- A linear system may behave in any one of three possible ways:
- For linear equations, logical independence is the same as linear independence.
- This is an example of equivalence in a system of linear equations.
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- Two kinds of equations are linear and quadratic.
- Linear equations can have one or more variables.
- Linear equations do not include exponents.
- An example of a graphed linear equation is presented below.
- (If $a=0$, the equation is a linear equation.)
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- A linear equation written in standard form makes it easy to calculate the zero, or $x$-intercept, of the equation.
- Standard form is another way of arranging a linear equation.
- In the standard form, a linear equation is written as:
- However, the zero of the equation is not immediately obvious when the linear equation is in this form.
- Convert linear equations to standard form and explain why it is useful to do so
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- A linear inequality is a mathematical statement that one linear expression is greater than or less than another linear expression.
- A linear equation, we know, may have exactly one solution, infinitely many solutions, or no solution.
- Speculate on the number of solutions of a linear inequality.
- A linear inequality may have infinitely many solutions or no solutions.
- Inequalities can be solved by basically the same methods as linear equations.