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One-Variable Equations Help

By — McGraw-Hill Professional
Updated on Aug 26, 2011

Elementary Rules

The objective of solving a single-variable equation is to get it into a form where the expression on the left-hand side of the equals sign is the variable being sought (for example, x) standing all alone, and the expression on the right-hand side of the equals sign is an expression that does not contain the variable being sought.

There are several ways in which an equation in one variable can be manipulated to obtain a solution, assuming a solution exists. The following rules can be applied in any order, and any number of times.

Addition of a quantity to each side: Any defined constant, variable, or expression can be added to both sides of an equation, and the result is equivalent to the original equation.

Subtraction of a quantity from each side: Any defined constant, variable, or expression can be subtracted from both sides of an equation, and the result is equivalent to the original equation.

Multiplication of each side by a quantity: Both sides of an equation can be multiplied by a defined constant, variable, or expression, and the result is equivalent to the original equation.

Division of each side by a quantity: Both sides of an equation can be divided by a nonzero constant, by a variable that cannot attain a value of zero, or by an expression that cannot attain a value of zero over the range of its variable(s), and the result is equivalent to the original equation.

Basic Equation and Factored Equations in One Variable

Basic Equation in One Variable

Consider an equation of the following form:

    ax + b = cx + d

where a, b, c, and d are real-number constants, x is a variable, and ac. This equation is solved for x as follows:

    ax + b = cx + d
    ax = cx + db
    axcx = db
    (ac)x = db
    x = (db)/(ac)

Factored Equations in One Variable

Consider an equation of the following form:

    (xa1)(xa2)(xa3) . . . (xan) = 0

where a1, a2, a3, . . ., an are real-number constants, and x is a variable. There are multiple solutions to this equation. Call the solutions x1, x2, x3, and so on up to xn, as follows:

      x1 = a1
      x2 = a2
      x3 = a3
          ↓
      xn = an

The solution set of this equation is {a1, a2, a3, . . ., an}.

Quadratic Equations

Consider an equation of the following form:

    ax2 + bx + c = 0

where a, b, and c are real-number constants, x is a variable, and a is not equal to 0. This is called the standard form of a quadratic equation. It may have no real-number solutions for x, or a single real-number solution, or two real-number solutions. The solutions of this equation, call them x1 and x2, can be found according to the following formulas:

    x1 = [–b + (b2 – 4ac)1/2]/2a
    x2 = [–b – (b2 – 4ac)1/2]/2a

Sometimes these are written together as a single formula, using a plus-orminus sign (±) to indicate that either addition or subtraction can be performed. This is the well-known quadratic formula from elementary algebra:

    x = [–b ± (b2 – 4ac)1/2]/2a
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