Linear B

Examples of Linear B in the following topics:

• Linear Equations

  • A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable.
  • A common form of a linear equation in the two variables $x$ and $y$ is:
  • where $m$ and $b$ designate constants.
  • In this particular equation, the constant $m$ determines the slope or gradient of that line, and the constant term $b$ determines the point at which the line crosses the $y$-axis, otherwise known as the $y$-intercept.
  • Linear differential equations are of the form:

• Linear and Quadratic Functions

  • In calculus and algebra, the term linear function refers to a function that satisfies the following two linearity properties:
  • Linear functions may be confused with affine functions.
  • One variable affine functions can be written as $f(x)=mx+b$.
  • However, the term "linear function" is quite often loosely used to include affine functions of the form $f(x)=mx+b$.
  • Linear functions form the basis of linear algebra.

• The Equation of a Line

  • In statistics, linear regression can be used to fit a predictive model to an observed data set of $y$ and $x$ values.
  • In statistics, simple linear regression is the least squares estimator of a linear regression model with a single explanatory variable.
  • A common form of a linear equation in the two variables $x$ and $y$ is:
  • Where $m$ (slope) and $b$ (intercept) designate constants.
  • In this particular equation, the constant $m$ determines the slope or gradient of that line, and the constant term $b$ determines the point at which the line crosses the $y$-axis, otherwise known as the $y$-intercept.

• Linear and Quadratic Equations

  • The constants $a$, $b$, and $c$ are respectively called the quadratic coefficient, the linear coefficient, and the constant term (or free term).
  • $\displaystyle x=\frac { -b\pm \sqrt { { b }^{ 2 }-4ac } }{ 2a }$
  • $\displaystyle x=\frac { -b + \sqrt { { b }^{ 2 }-4ac } }{ 2a }$
  • $\displaystyle x=\frac { -b- \sqrt { { b }^{ 2 }-4ac } }{ 2a }$
  • Graph sample of linear equations, using the y=mx+b format, as seen by $y=-x+5$(red) and $y=\frac{1}{2}x +2$ (blue).

• Slope and Y-Intercept of a Linear Equation

  • For the linear equation y = a + bx, b = slope and a = y-intercept.
  • A linear equation that expresses the total amount of money Svetlana earns for each session she tutors is y = 25 + 15x.
  • The slope is 15 (b = 15).
  • (a) If b > 0, the line slopes upward to the right.
  • (b) If b = 0, the line is horizontal.

• Linear Equations

  • Linear regression for two variables is based on a linear equation with one independent variable.
  • The graph of a linear equation of the form y = a + bx is a straight line.
  • Linear equations of this form occur in applications of life sciences, social sciences, psychology, business, economics, physical sciences, mathematics, and other areas.

• What is a Linear Function?

  • For example, a common equation, $y=mx+b$, (namely the slope-intercept form, which we will learn more about later) is a linear function because it meets both criteria with $x$ and $y$ as variables and $m$ and $b$ as constants.  
  • In the linear function graphs below, the constant, $m$, determines the slope or gradient of that line, and the constant term, $b$, determines the point at which the line crosses the $y$-axis, otherwise known as the $y$-intercept.
  • Horizontal lines have a slope of zero and is represented by the form, $y=b$, where $b$ is the $y$-intercept.  
  • 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

• Solving Problems with Inequalities

  • A linear inequality is a mathematical statement that one linear expression is greater than or less than another linear expression.
  • Let a, b, and c represent real numbers and assume that a < b.
  • a + c < b + c and a − c < b − c.
  • If c is a positive real number, consider then if a < b, ac < bc and ac < bc.
  • If c is a negative real number, then if a < b, ac > bc and ac > bc.

• Linear Approximation

  • A linear approximation is an approximation of a general function using a linear function.
  • In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function).
  • Linear approximations are widely used to solve (or approximate solutions to) equations.
  • Linear approximation is achieved by using Taylor's theorem to approximate the value of a function at a point.
  • If one were to take an infinitesimally small step size for $a$, the linear approximation would exactly match the function.

• Linear Equations and Their Applications

  • 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:
  • where a, b, c, and d are constants and x, y, and z are variables.
  • Imagine these linear equations represent the trajectories of two vehicles.