linear equation

Examples of linear equation in the following topics:

• 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.

• 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.
  • For example, consider the following system of linear equations in two variables:
  • The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently.
  • In this example, the ordered pair (4, 7) is the solution to the system of linear equations.
  • Note that a system of linear equations may contain more than two equations, and more than two variables.

• 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

• 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.)

• Inconsistent and Dependent Systems

  • In mathematics, a system of linear equations (or linear system) is a collection of linear equations involving the same set of variables.
  • The equations of a linear system are independent if none of the equations can be derived algebraically from the others.
  • For linear equations, logical independence is the same as linear independence.
  • This is an example of equivalence in a system of linear equations.
  • It is possible for three linear equations to be inconsistent, even though any two of them are consistent together.

• Matrix Equations

  • Matrices can be used to compactly write and work with systems of multiple linear equations.
  • This is very helpful when we start to work with systems of equations.
  • Solving a system of linear equations using the inverse of a matrix requires the definition of two new matrices: $X$ is the matrix representing the variables of the system, and $B$ is the matrix representing the constants.
  • Using matrix multiplication, we may define a system of equations with the same number of equations as variables as:
  • To solve a system of linear equations using an inverse matrix, let $A$ be the coefficient matrix, let $X$ be the variable matrix, and let $B$ be the constant matrix.

• Linear Inequalities

  • 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.  
  • The same properties for solving an equation are used to solve an inequality; however, when solving an equation there is one solution (or one value that makes the equation true), but when solving an inequality there are many solutions (or values that make the statement true).
  • Solving the inequality is the same as solving an equation.  
  • Step 3, this is similar to solving a two step equation.

• Solving Systems of Equations in Three Variables

  • In mathematics, simultaneous equations are a set of equations containing multiple variables.
  • This is a set of linear equations, also known as a linear system of equations, in three variables:
  • Elimination by judicious multiplication is the other commonly-used method to solve simultaneous linear equations.
  • Now subtract two times the first equation from the third equation to get
  • Finally, subtract the third and second equation from the first equation to get

• Solving Systems Graphically

  • A simple way to solve a system of equations is to look for the intersecting point or points of the equations.
  • A system of equations (also known as simultaneous equations) is a set of equations with multiple variables, solved when the values of all variables simultaneously satisfy all of the equations.
  • In a set of linear equations (such as in the image below), there is only one solution.
  • Once you have converted the equations into slope-intercept form, you can graph the equations.
  • This graph shows a system of equations with two variables and only one set of answers that satisfies both equations.

• 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.
  • 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.  
  • 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.
  • Identify what makes a function linear and the characteristics of a linear function