What Is an Equation?

An equation shows a relationship between two quantities. Here are examples of equations:

It’s all about the relationship between two sides of the equation.

An equation has two sides, separated by an equals sign. Think of a seesaw and how two people one either side are needed to achieve balance.

True and False Equations

An equation can be true or false. A true equation is mathematically accurate. These are examples of true equations:

A false equation is mathematically inaccurate. Here are examples of false equations.

If an equation has variables, it is a true equation when the variable equals the solution(s) to the equation

The Reflexive Property of Equality.

The simplest true equation is when a quantity is equal to itself. For example,

This is an example of the Reflexive Property of Equality. You can always write an equation by making any expression equal to itself. For the equation shown above, not only is it a true equation, it is true for any value of x. An equation that is true for any value of the variable is also known as an identity.

Conditional Equations

Most of the equations that you’ll deal with are called conditional equations. Conditional equations are true but only for certain values of the variable. In fact, solving an equation is finding the conditions under which the equation is true.

Let’s look at an example.

This equation isn’t true for all values of x. For example, look at the equation when x = 0.

Under what conditions is this equation true?

By solving this equation, you will find the conditions under which this equation is true. Let’s solve it.

$<math xmlns="http://www.w3.org/1998/Math/MathML" xmlns:wrs="http://www.wiris.com/xml/mathml-extension"><mtable wrs:columnalign="center relation left"><mtr><mtd><mi>S</mi><mi>t</mi><mi>e</mi><mi>p</mi></mtd><mtd><mi>E</mi><mi>q</mi><mi>u</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></mtd><mtd><mi>D</mi><mi>e</mi><mi>s</mi><mi>c</mi><mi>r</mi><mi>i</mi><mi>p</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>3</mn><mi>x</mi><mo>+</mo><mn>5</mn><mo>=</mo><mn>20</mn></mtd><mtd><mi>I</mi><mi>n</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>a</mi><mi>l</mi><mo> </mo><mi>e</mi><mi>q</mi><mi>u</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd><mtd><mn>3</mn><mi>x</mi><mo>+</mo><mn>5</mn><mo>-</mo><mn>5</mn><mo>=</mo><mn>20</mn><mo>-</mo><mn>5</mn></mtd><mtd><mi>S</mi><mi>u</mi><mi>b</mi><mi>t</mi><mi>r</mi><mi>a</mi><mi>c</mi><mi>t</mi><mo> </mo><mn>5</mn><mo> </mo><mi>f</mi><mi>r</mi><mi>o</mi><mi>m</mi><mo> </mo><mi>b</mi><mi>o</mi><mi>t</mi><mi>h</mi><mo> </mo><mi>s</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>s</mi><mo> </mo><mi>o</mi><mi>f</mi><mo> </mo><mi>t</mi><mi>h</mi><mi>e</mi><mo> </mo><mi>e</mi><mi>q</mi><mi>u</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo>.</mo></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mn>3</mn><mi>x</mi><mo>=</mo><mn>15</mn></mtd><mtd><mi>S</mi><mi>i</mi><mi>m</mi><mi>p</mi><mi>l</mi><mi>i</mi><mi>f</mi><mi>y</mi><mo>.</mo></mtd></mtr><mtr><mtd><mn>4</mn></mtd><mtd><mfrac><mrow><mn>3</mn><mi>x</mi></mrow><mn>3</mn></mfrac><mo>=</mo><mfrac><mn>15</mn><mn>3</mn></mfrac></mtd><mtd><mi>D</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>d</mi><mi>e</mi><mo> </mo><mi>b</mi><mi>o</mi><mi>t</mi><mi>h</mi><mo> </mo><mi>s</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>s</mi><mo> </mo><mi>o</mi><mi>f</mi><mo> </mo><mi>t</mi><mi>h</mi><mi>e</mi><mo> </mo><mi>e</mi><mi>q</mi><mi>u</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo> </mo><mi>b</mi><mi>y</mi><mo> </mo><mn>3</mn><mo>.</mo></mtd></mtr><mtr><mtd><mn>5</mn></mtd><mtd><mi>x</mi><mo>=</mo><mn>5</mn></mtd><mtd><mi>S</mi><mi>o</mi><mi>l</mi><mi>u</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></mtd></mtr></mtable></math>$

So, the conditional equation is a true equation only for x = 5. For all other values of x, the equation is false.