Simple Equations

< 28.8 Visualizing Algebraic Expressions 2 | Topic Index | 28.10 Solving Simple Equations >

Algebraic expressions are made of variables &amp; arithmetic operations. Variables can take numerical values which in turn give numerical values to the expressions.

Value of an Algebraic Expression 

We saw that the expression x + 32 could convert any temperature given in Centigrade scale into an equivalent temperature in the Fahrenheit scale. The following table shows the temperature in Centigrade converted to its equivalent in Fahrenheit

We can see that for different values of the variable, the expression takes a particular value.

Equation

The idea behind equations is to do the reverse of the above process. In equations, the value of the expression is given and we have to find the value of the variable for which it is true. For example if the value of the expression (temperature in F) is given as 68, then the problem is written in the following format.

x + 32 = 68

The aim is to find that value of &ldquo;x&rdquo; for which the expression &ldquo;x + 32&rdquo; has a value of 68. This is called &ldquo;solving&rdquo; the equation.

The two sides of the equation are called the &ldquo;Left side&rdquo; and &ldquo;right side&rdquo; of the equation. The &ldquo;equal&rdquo; sign &ldquo;=&rdquo; shows that both the sides have the same value.

What we have shown here is one of the simplest types of equations. It is called a &ldquo;linear equation of a single variable&rdquo;. In higher classes we will study many other types of equations.

The term &ldquo;linear&rdquo; has some relation to the equivalence between a line and an expression of degree 1, which we saw in the previous chapter. The meaning of the term &ldquo;single variable&rdquo; is obvious.

Visualizing an Equation

One of the ways to understand and equation and its solution is to visualize an equation as weighing balance with 2 pans on either side of a fulcrum.

Power of Algebraic language

An entire infinity of addition relations between 3 numbers (a, b &amp; c) can now be expressed as &ldquo;x + y = z&rdquo;. Not only that. A series of related relations can also be expressed.

Hence &ldquo;x + y = z&rdquo; also implies: &ldquo;y + x = z&rdquo;, &ldquo;z – y = x&rdquo; &amp; &ldquo;z – x = y&rdquo;.

The greatest advantage of algebra is that the various mathematical operations can be performed without the cumbersome attachments of concrete entities&mdash;entities like dollars, angles, area, percentages, groupings of pencils. Once a particular word problem has been translated into a mathematical representation, the entirety of its mathematically relevant content is condensed onto abstract symbols, freeing working memory and unleashing the power of pure mathematics. With working memory less burdened, the student can focus on solving the problem at hand.

For example representing the area of a rectangle as A=L*B and the perimeter as P=2*(L + B) allows us to represent the relation between A &amp; P as another algebraic expression and explore the relation between area &amp; perimeter of a rectangle and find interesting relations.

< 28.8 Visualizing Algebraic Expressions 2 | Topic Index | 28.10 Solving Simple Equations >