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.

• 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.

• 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.

• What is an Equation?

  • In an equation with one variable, the variable has a solution, or value, that makes the equation true.
  • In many cases, an equation contains one or more variables.
  • It is possible for equations to have more than one variable.
  • The values of the variables that make an equation true are called the solutions of the equation.
  • In turn, solving an equation means determining what values for the variables make the equation a true statement.

• Linear and Quadratic Equations

  • In an equation with a single unknown, a value of that unknown for which the equation is true is called a solution or root of the equation.
  • In a set simultaneous equations, or system of equations, multiple equations are given with multiple unknowns.
  • Linear equations do not include exponents.
  • A quadratic equation is a univariate polynomial equation of the second degree.
  • (If $a=0$, the equation is a linear equation.)

• Inconsistent and Dependent Systems in Three Variables

  • The three planes could be the same, so that a solution to one equation will be the solution to the other two equations.
  • First, multiply the first equation by $-2$ and add it to the second equation:
  • Just as with systems of equations in two variables, we may come across an inconsistent system of equations in three variables, which means that it does not have a solution that satisfies all three equations.
  • Next, multiply the first equation by $-5$,  and add it to the third equation:
  • We can solve this by multiplying the top equation by 2, and adding it to the bottom equation:

• Solving Equations: Addition and Multiplication Properties of Equality

  • In an equation with a single unknown, a value of that unknown for which the equation is true is called a solution or root of the equation.
  • If an equation in algebra is known to be true, the following properties may be used to produce another true equation.
  • In other words, any real number can be added to both sides of an equation.
  • The equation is therefore:
  • First, use the addition property to add 5 to both sides of the equation: