linear

Examples of linear in the following topics:

• Zeroes of Linear Functions

  • 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 $x$.

• 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:
  • For example,imagine these linear equations represent the trajectories of two vehicles.
  • Imagine these linear equations represent the trajectories of two vehicles.

• Linear Inequalities

  • 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.  

• Introduction to Systems of Equations

  • 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:

• What is a Linear Function?

  • 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

• Inconsistent and Dependent Systems

  • 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.

• Linear and Quadratic Equations

  • 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.)

• Linear Equations in Standard Form

  • 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

• Solving Problems with Inequalities

  • 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.

• Formulas and Problem-Solving

  • Linear equations can be used to solve many everyday and technically specific problems.
  • Linear equations can be used to solve many practical and technical problems.
  • For example, one can use a linear equation to determine the amount of interest accrued on a home equity line of credit after a given amount of time.
  • Let's take a few examples of other linear equations, namely velocity, gratuity (tip), and cost of purchased goods:
  • Use a given linear formula to solve for a missing variable