Earlier we learnt about variables and expressions . In this chapter we will learn some simple techniques to write correct equations for some given problem.
We saw that an algebraic expression is formed using coefficients combined with variables raised to different powers.
Equations can be formed by combining exactly two expressions using the equal to $(=)$ sign.
$Expression1 = Expression2$
Following are some examples of equations:
$4x^2 + 3xy - 46y^2 = 5x^2y + 7x^2y$
$11x^2 - 17x + 22 = 10x + 5$
$13x + 20 = 7x + 12$
One of the two expressions can be a single constant term like:
$3x + 17 = 32$
$10x^2 + 15x = 22$
$17x - 51 = 0$
-----------book page break-----------
Now let us try the following problem:
--------- Reference to question: f12aa09d-eed3-4d29-9dfc-62bac22ff0cc ---------
-----------book page break-----------
II. Linear Equation In Single Variable
The equations we saw in the previous section, were a mix of linear and non-linear equations in one or more variables.
In this section and the rest of the chapter we will focus on the linear equations in a single variable.
Any equation, in one or more variables represents a straight line when the maximum power of any of the terms in the equation is no more than $1$.
For example the equation:
$3x + 11 = 23$ represents a straight line in one dimension.
The equation:
$3x + 7y = 27$ represents a straight line in
two dimensions
.
The equation:
$11x + 6y + 18z = 29$ represents a straight plane in
three dimensions
.
However, the equation
$x^2 + y^2 = 36$ does not represent a straight line but a circle in two dimensions.
Linear equations are equations where the degree of every variable term is exactly $1$.
Now, try this problem:
-----------book page break-----------
--------- Reference to question: 8bc20971-badd-49ec-b0bd-ab818815078d ---------
-----------book page break-----------
III. Forming Equations
Writing equation is a simple process of converting a mathematical problem expressed in plain natural language like English, to mathematical language, using the signs, symbols and numbers.
Let us look at the statement below:
$\text{If you add 10 to 3 times A's current age, you get four times his current age.}$
Here although the value of $A$'s age has not been given, we do have some information about $A$'s age. Let us see if we can form an equation with whatever information has been provided here or not.
Let us start by forming the first expression.
If we use $x$ to represent $A$'s age:
Three times $A$'s current age is
$3x$
Adding $10$ to $3$ times $A's$ age gives us
$3x + 10$
Like this we can form the second expression.
Four times $A$'s age is represented by
$4x$
The problem says that these two quantities are equal, therefore we get our equation as:
$3x + 10 = 4x$
-----------book page break-----------
Now let us try the following problem:
--------- Reference to question: 59f3611b-fdf9-4849-bf99-06a2070a9c17 ---------
In the next section we will look at how to solve linear equations in one variable.
-----------book page break-----------
IV. Solving Equations
Given any equation in a single variable $x$ our aim is to rearrange the terms of the equation such that we get an equation of the form:
$x = V$ where $V$ is a numeric value and does not contain any variable.
At this stage we can say that we have found the value of the unknown $x$.
Rules for solving:
Since both sides of an equation represent equal numbers, performing the following steps will not change the equality:
$\cdot \text{ Adding the same value to both sides of the equal to sign.}$
$\cdot \text{ Subtracting the same value from both sides of the equation.}$
$\cdot \text{ Multiplying both sides by the same value.}$
$\cdot \text{ Dividing both sides by the same value, except zero.}$
We can also perform any other mathematical operation on both sides, like raising both sides to some power, or taking logarithm or applying any trigonometric function to both sides. But for now we will focus on solving the equation using the first four steps only.
Let us try to solve the equation we formed for $A$'s age in the previous section.
We got our equation as:
$3x + 10 = 4x$
As a first step we will subtract $3x$ from both sides, which will give us:
$3x + 10 - 3x = 4x - 3x$
$\Rightarrow 10 = x$
Now, if $10 = x$ the we can interchange the values on both sides to write $x = 10$
So, we get our final answer as $A$'s current age is $10$.
-----------book page break-----------
Observe, that while we went from one step to the next one we used the symbol $\Rightarrow$. This symbol means $follows\ that$ or $implies$, which means that this step follows from the previous step.
So, when we say:
$3x + 10 = 4x$
$\Rightarrow x = 10$
It $DOES\ NOT$ mean that $3x + 10$ is equal to $x$ or $4x$ is equal to $10$.
It means that $3x + 10 = 4x$ implies that $x = 10$
For example,
$6y = 30$
$\Rightarrow y = 5$
means that the statement $6y = 30$ implies that $y = 5$
Now let us try the following problem:
--------- Reference to question: a7e62dc7-a588-4ca7-8104-b008bdbf9bd2 ---------