3. Equations & Inequalities

Lesson

Out in the real world, all sorts of amazing relationships play out every day: Air temperatures change with ocean temperatures. Populations of species rise and fall depending on seasons, food availability and the number of predators. The surface area of a human body can even be measured fairly accurately according to your height and weight.

One of the most powerful things about mathematics is its ability to describe and measure these patterns and relationships exactly. Given a mathematical formula for the relationship between, say, the weight of a patient and how much medication they should be given, we can find one quantity by substituting a value for the other.

We have come across so many different formulas in mathematics that allow us to measure quantities such as Area, Volume, Speed etc. Let's have a look at the process of substituting values into these formulas to find a particular unknown.

The perimeter of a triangle is defined by the formula $P=x+y+z$`P`=`x`+`y`+`z`. Find $P$`P` if the length of each of its three sides are $x=5$`x`=5 cm, $y=6$`y`=6 cm and $z=3$`z`=3 cm.

**Solution**:

By inserting the number values of $x$`x`, $y$`y` and $z$`z` we have a new equation that we can use to find the value of $P$`P`:

$P=5+6+3$`P`=5+6+3

$P=14$`P`=14 cm

The area of a square with side $a$`a` is given by the formula $A=a^2$`A`=`a`2. Find $A$`A` if $a=6$`a`=6 cm.

**Solution**:

From the information above, we know that we are finding the area of a square where each side measures $6$6cm.

Substituting our value for $a$`a` into the formula:

$A=6^2$`A`=62

$A=36$`A`=36 $cm^2$`c``m`2

The simple interest generated by an investment is given by the formula $I=\frac{P\times R\times T}{100}$`I`=`P`×`R`×`T`100.

Given that $P=1000$`P`=1000, $R=6$`R`=6 and $T=7$`T`=7, find the interest generated.

The surface area of a rectangular prism is given by formula $S=2\left(lw+wh+lh\right)$`S`=2(`l``w`+`w``h`+`l``h`), where $l$`l` , $w$`w` and $h$`h` are the dimensions of the prism.

Given that a rectangular prism has a length of $8$8 cm, a width of $7$7 cm and a height of $9$9 cm, find its surface area.

Solving for a quantity of interest in a literal equation is an important skill to learn. It can come in very handy when you know the value of one algebraic symbol but not another.

For example, in the formula $A=pb+y$`A`=`p``b`+`y`, the value $A$`A` is by itself on the left-hand side of the equals sign. In common language, we might say that it has been "solved for" because it is by itself, even though we do not yet know its value.

When we previously tried to solve equations, we took steps to get the variable by itself. When solving for a quantity of interest, we might have more than one variable, but we still use a similar process:

- Group any like terms
- Simplify using the inverse of addition or subtraction.
- Simplify further by using the inverse of multiplication or division.

Solve for $x$`x` in the following equation:

$y=\frac{x}{4}$`y`=`x`4

Solve for $R$`R` in the following equation:

$V=IR-E$`V`=`I``R`−`E`

Solve for $x$`x` in the following equation:

$\frac{x}{9}+\frac{n}{2}=5$`x`9+`n`2=5