Formulas and Equations

An equation is a mathematical sentence that contains one or more variables and one or more mathematical operators (such as +, −, ÷, ×, and =). An equation expresses a relationship between two or more quantities.

A formula is a special kind of equation. A formula such as $V\mathit{ }\mathit{=}\mathit{ }l\mathit{ }\mathit{\times }\mathit{ }w\mathit{ }\mathit{\times }\mathit{ }h$ states the relationship between unknown quantities represented by the variables V, l, w, and h. The formula means that volume (of a rectangular solid) equals length times width times height. Some formulas have numbers that do not vary, such as the formula for the perimeter of a square: P = 4s. In this formula, the number 4 is a constant.

To solve an equation or formula for an unknown quantity, first rearrange the equation so that the unknown is on one side of the equation, and all the known quantities are on the other side. Then substitute known values for the variables. Be sure to include units.

Example

An airplane travels in a straight line at a speed of 600 km/h. How far does it fly in 3.5 hours?

Write the formula that relates speed, distance, and time.

To solve for distance, multiply both sides of the equation by t.

Substitute in the known values.

$600km/h\times 3.5h=d2100km=d$

Example

What is the volume of 642 g of gold if the density of gold is 19.3 g/cm³?

Write the formula that relates density, mass, and volume.

First solve the equation for the unknown quantity, volume (v).

Then substitute in the known values for m and d.

$v=\genfrac{}{}{0.1ex}{}{642g}{19.3g/c{m}^3}v=33.2645c{m}^{3}v=33.3c{m}^3$

Example

A gas has a volume of 5.0 L at a temperature of 200 K. The temperature of the gas is increased under constant pressure until the final volume of the gas is 15 L. What is the final temperature of the gas?

Write the formula that describes how the volume of a gas changes with temperature if the pressure and number of particles are constant.

$\genfrac{}{}{0.1ex}{}{V_1}{T_1}=\genfrac{}{}{0.1ex}{}{V_2}{T_2}$

First solve the equation for the unknown quantity, T₂.

$\genfrac{}{}{0.1ex}{}{V_1}{T_1}=\genfrac{}{}{0.1ex}{}{V_2}{T_2}{T}_2=\genfrac{}{}{0.1ex}{}{V_2\times T_1}{V_1}$

Now substitute the known values for V₁, V₂, and T₁.

$T_2=\genfrac{}{}{0.1ex}{}{15L\times 200K}{5L}{T}_2=600K$

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