An equation is a statement of equality. Should this statement involve a variable it will, unless it is an invalid equation, be true for one or more values of that variable, though these values need not be expressible in terms of real numbers: x ² + 2 = 0 has two imaginary (see imaginary numbers) roots, +√–2 and –√–2.

 

Linear equations

Linear equations are those in which no variable term is raised to a power higher than 1. Solution of linear equations in one variable is simple. Consider the equation x + 3 = 7. The equation will still be true if we add or subtract equal numbers from each side:

 

    x + 3 = 7
    x + 3 – 3 = 7 – 3
    and x = 4.

 

Linear equations are so called because, if considered as the equivalent of a curve, they can be plotted as a straight line.

 

Quadratic equations

Quadratic equations are those in a single variable which appears to the power 2, but not higher. A quadratic equation always has two roots though these roots may be equal.

 

Cubic equations

Cubic equations are those in a single variable which appears to the power 3, but not higher. Cubic equations always have three roots, though two or all three of these may be equal.

 

Degree of an equation

Linear, quadratic, and cubic equations are said to be of the 1st, 2nd, and 3rd degrees respectively. More generally, the degree of an equation is defined as the sum of the exponents of the variables in the highest-power term of the equation. In ax ⁵ + bx ³y ³ + cx ²y ⁵ = 0, the sums of the exponents of each term are, respectively, 5, 6, and 7; hence cx ²y ⁵ is the high test power term, and the equation is of the 7th degree.

 

Radical equations

Radical equations are those in which the roots of the variables appear: e.g., a ^p√ + b ^q√ + c = 0. Radical equations can always be simply converted into equations of the nth order, where n = 1, 2, 3 ..., by raising both sides of the equation to a power, repeating the process where necessary.

 

Simultaneous equations

A single equation in two or more variables is generally insoluble. However, if there are as many equations as there are variables, it is possible to solve for each variable. Consider:

 

    2x + xy + 3 = 0    (1)
    and x + 2xy = 0   (2)

 

Multiplying equation (1) by 2 we have

 

    4x + 2xy + 6 = 0   (3)

 

and subtracting equation (2) from this,

 

    3x + 6 = 0. Hence x = –2.

 

Substituting this value into equation (1) we find the value y = –½. More complicated simultaneous equations can be solved in the same way.