9. Linear Equations & Inequalities
A mathematical statement with an equal to sign in it, with a left hand side and right hand side parts (either side of the "=") is called an equation.
e.g. 2+3=5, here LHS is 2+3, RHS is 5, and the statement says LHS=RHS and in this case this happens to be true
A similar statement where there is no "=" sign, but the equation is split into LHS and RHS parts by > or < is called an inequality
e.g. 2+3 < 6, where LHS is 2+3, RHS is 6, and since 5 < 6, this inequality is true.
An equation or inequality that has letters to "stand in" for numbers, is called an algebraic expression.
The letters that stand in for numbers are called variables.
e.g. 2a+3b=5 here LHS=2a+3b where a and b are variables, RHS is 5, and we see if a=1 and b=1, then LHS=2(1)+3(1)=2+3=5=RHS
The process of finding the values of the variables that "satisfies" or makes true the statement, is called solving.
e.g. solve: $2x+4=17$
Subtracting 4 from both sides of the equation (to retain equality), we get: $2x=17-4=13$.
Dividing both sides by 2, we get $(2x)/2=13/2$ or $x=13/2$. This is our answer.
e.g. solve: $2x+4 < 17$
Proceeding as above, we get: $2x < 13$ or $x < 13/2$
Important: for inequalities we keep the sign the same for all operations except when we divide or multiply by a negative number, when we flip it.
The highest power of a variable in an equation is the degree of the equation.
A linear equation (or inequality) has degree 1.
e.g. $2x+3y=7$ is a linear equation because the exponents of variables x and y are both 1.
e.g. $ax^2+bx+c=0$ where a, b, and c are constants, and x is a variable, has degree 2, which is the largest exponent of the variable x.
The second example above is called a quadratic equation. It is not linear.
When solving an equation (or inequality), the variables whose values are at first unknown, need to be determined.
We need at least as many equations as there are unknown variables, in order to solve them for a unique value of a variable
e.g. $x+2y=3$ cannot be solved for a unique value of x or y unless additional constraints are imposed.
e.g. $x+2y=3$ and $2x+y=3$ together can be solved (2 equations with 2 unknowns, x and y) for unique values of x and y.
In the second example above, we are solving a "system of equations" to determine the values of x and y.
Please note that it is important that the equations in a system need to be independent if we are to solve them.
e.g. $x+2y=3$ and $2x+4y=6$ cannot be solved for a unique set of values for x and y because the second equation is a simple multiple of the first.
The two equations in this case are not independent, so the system cannot be solved for unique values x and y.