Concept of Linear Equations and Modeling

Basic Definitions of Linear Equations and Modeling:

Equation: It is a statement in which two expressions are equal.

Solutions: These are the values that make the equation true.

Identity: An equation that is true for each and every real number in the domain.

Contradiction: An equation that is false for each and every real number in the domain.

Conditional equation: An equation that might be false or true depending on the values of variables.
Equivalent equations: These are the equations having similar solution set.

Linear equation in one variable: Equation which can be written as ax + b  = 0, where a and b are real and a does not equivalent to zero. If did equivalent to zero, it would be a constant equation and an identity when b = 0 or a contradiction when b ≠ 0.

Extraneous solutions: These are the Solutions that satisfy an ‘equivalent’ equation, however not the original equation. It can be introduced by dividing or multiplying by an expression containing a variable. They can as well be introduced by applying a non-one-to-one function to both sides (such as squaring both sides). You should always check your outcome or answer when there is a possibility that you have introduced an extraneous solution.

Mathematical model: It is an algebraic equation used to resolve a problem which occurs in real life.

Equivalent Operations:

The given below operations can be employed to produce an equivalent equation:

a) Eliminate grouping symbols.

b) Join similar terms.

c) Reduce the fractions.

d) Add similar quantity to both sides of the equation.

e) Subtract the similar quantity from both sides of the equation.

f) Multiply both sides of the equation by similar non-zero quantity.

g) Divide both sides of the equation by similar non-zero quantity.

h) Apply one-to-one function to both sides of equation. Be cautious of domains whenever doing this, although.

Solving Equations:

There are some general guidelines, individual problems might differ.

Linear Equations:

a) Remove any grouping symbols.

b) Use addition or subtraction to move all the terms having the variable to one side and all terms devoid of the variable to other side. Simplify if there is more than one occurrence of variable, you might require factoring the variable out.

c) Use multiplication or division to obtain the variable by itself.

d) Check your outcome or answer.


a) Multiply each and every term of both sides of the equation by the power of 10 (10, 100, 1000, and so on) to remove the decimals. Note that the expressions such as 0.42 (x + 2) are just one term since of multiplication. Either multiply 0.42 by 100, or (x+2) by 100 (ok, but it does not get rid of decimals), however not 0.42 by 100 and (x+2) by 100 (you have really multiplied by 10,000 if you do that).

b) Finish the problem as it was the regular linear equation.

Note that there is no need to remove the decimals. Several people prefer to work with the decimals. And other people find it harder. This is a personal preference; however most of the people will remove the decimals.


The fractions are your friends. They will hang out with you on weekends when no one else will. Though, most of the people find it hard to work with fractions, and though they are your friends, solving equations becomes quite simpler when they are not around.

a) Find out the least common denominator out of all the denominators. Be sure to factor any denominator first whenever possible.

b) Note anywhere that the values that make the LCD zero cannot be utilized. LCD is a collection of all things in the denominator, and therefore anything which makes it zero would cause division by zero and this is an extremely bad thing. Our main goal is to get rid of fractions, therefore in the next step there will be no more denominator. That signifies that what was in implied domain is no longer in the implied domain and should be stated overtly.

Multiply each and every term of both sides of the equation by the LCD (least common denominator). All the denominators will inverse out and become 1 (and therefore do not require to be written).

Note that if you're working with an equation, the fractions can be eliminated. No such claim can be made about working with expressions, however. It is the multiplicative property of equality which allows us to eliminate the denominators, and you just don't have equality when you're working with expressions.

Mathematical Models:

Mathematical modeling is the procedure of taking a verbal explanation of a problem, assigning labels to unknown quantities and making a mathematical model or the algebraic equation.


Given below are some words and what they generally mean.

Equality: Represents, is, are, will be.

Addition: Sum, more than, plus, increased by, greater, exceeds, total

Subtraction: Difference, minus, less, subtracted from, decreased by, reduced by, the remainder

Multiplication: Product, twice, multiplied by, times, percent of

Division: Quotient, ratio, divided by, per


There are some general formulas, formulas from geometry, and conversions.

You can avoid the formulas dealing with trigonometric functions (Sin, Cos and Tan) and the region of a sector. You do not require to memorize, however do not forget that they are there, the formulas for equilateral triangle, circular ring, right circular cone and frustrum.

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