equation

Examples of equation in the following topics:

• 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.
  • When the equations are independent, each equation contains new information about the variables, and removing any of the equations increases the size of the solution set.
  • For example, the equations
  • Adding the first two equations together gives 3x + 2y = 2, which can be subtracted from the third equation to yield 0 = 1.

• 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.
  • Once you have converted the equations into slope-intercept form, you can graph the equations.
  • To determine the solutions of the set of equations, identify the points of intersection between the graphed equations.
  • This graph shows a system of equations with two variables and only one set of answers that satisfies both equations.

• Parametric Equations

  • Parametric equations are a set of equations in which the coordinates (e.g., $x$ and $y$) are expressed in terms of a single third parameter.
  • Converting a set of parametric equations to a single equation involves eliminating the variable from the simultaneous equations.
  • If one of these equations can be solved for $t$, the expression obtained can be substituted into the other equation to obtain an equation involving $x$ and $y$ only.
  • In some cases there is no single equation in closed form that is equivalent to the parametric equations.
  • One example of a sketch defined by parametric equations.

• 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:
  • The parametric form of a linear equation involves two simultaneous equations in terms of a variable parameter $t$, with the following values:
  • Linear differential equations are differential equations that have solutions which can be added together to form other solutions.
  • Linear differential equations are of the form:

• Solving Differential Equations

  • Differential equations are solved by finding the function for which the equation holds true.
  • A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself to its derivatives of various orders.
  • As you can see, such an equation relates a function $f(x)$ to its derivative.
  • Solving the differential equation means solving for the function $f(x)$.
  • The "order" of a differential equation depends on the derivative of the highest order in the equation.

• The Substitution Method

  • The substitution method is a way of solving a system of equations by expressing the equations in terms of only one variable.
  • The substitution method for solving systems of equations is a way to simplify the system of equations by expressing one variable in terms of another, thus removing one variable from an equation.
  • When the resulting simplified equation has only one variable to work with, the equation becomes solvable.
  • Note that now this equation only has one variable (y).
  • We can then simplify this equation and solve for y:

• 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:
  • Now subtract two times the first equation from the third equation to get
  • Next, subtract two times the third equation from the second equation and simplify:
  • Finally, subtract the third and second equation from the first equation to get

• Introduction to Systems of Equations

  • A system of equations consists of two or more equations with two or more variables, where any solution must satisfy all of the equations in the system at the same time.
  • 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.
  • The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently.
  • We can verify the solution by substituting the values into each equation to see if the ordered pair satisfies both equations.
  • Note that a system of linear equations may contain more than two equations, and more than two variables.

• Nonhomogeneous Linear Equations

  • When $f(t)=0$, the equations are called homogeneous second-order linear differential equations.
  • Otherwise, the equations are called nonhomogeneous equations.
  • Linear differential equations are differential equations that have solutions which can be added together to form other solutions.
  • This can be confirmed by substituting $y(x) = c_1y_1(t) + c_2 y_2(t)$ into the equation and using the fact that both $y_1(t)$ and $y_2(t)$ are solutions of the equation.
  • Identify when a second-order linear differential equation can be solved analytically

• 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.
  • In the standard form, a linear equation is written as:
  • The graph of the equation is a straight line, and every straight line can be represented by an equation in the standard form.
  • We know that the y-intercept of a linear equation can easily be found by putting the equation in slope-intercept form.
  • However, the zero of the equation is not immediately obvious when the linear equation is in this form.