Non Gamstop CasinosUK Non Gamstop CasinosBetting Sites That Are Not On GamstopCasino Non AamsCasino Non AamsCasino Italiani Non Aams

Equations and Inequalities

What you will learn in this lesson on Equations?

Equations are statements of equality.

x + 1 = 2 is a statement of equality between x + 1 and 2.

The statement is:

For what value of x, will x + 1 become equal to 2 ?

Answer: for x = 1.

Replacing x by 1 makes x + 1 equal to 2,

i.e. 1 + 1 = 2.

We say, x = 1 is solution of the equation:

x + 1 = 2.

Replacing x by 1 is called substitution

In the equation x + 1 = 2:   

x + 1 is said to be in the Left Hand Side, in short LHS, and 2, in the Right Hand Side, abbreviated as RHS

x is called  variable,  i,e what changes or varies.

The values of x for which the LHS and the RHS become equal are called Solution Set of the Equation.  

We also say solution of x satisfies the equation.

To satisfy an equation means the values of x which make LHS and RHS equal.

“To solve for x” is to find the solution of an equation.

An equation in which the highest exponent of the variable is 1 is called a linear equation.

In x + 1 = 2, the highest exponent of the variable x is 1.

So, we say x + 1 = 2 is a linear equation in variable x.

Other examples of linear equations are:

x + 3 = 5, in which the highest exponent of the variable x is 1

p + 9 = 2, in which the highest exponent of the variable p is 1

a + b = 1, in which the highest exponents of the two variables a and b are 1

Equations in which the highest exponent of the variable is 2 are called quadratic equations .

Examples of quadratic equations are:

x2 + 2x + 1 = 0, in which the highest exponent of the variable x is 2

x2 – 4 = 0, in which the highest exponent of the variable x is 2

y2 – 4ax = 0, in which the highest exponent of the variable y is 2

6p2 – p - 1 =0, in which the highest exponent of the variable p is 2.

If you wish to set off with your lesson on Equations, then click on any of the links below:

  • Linear Equation in One Variable:

  • Solution of an Equation:  

  • How to Find the Solution of Linear Equations in One Variable:

  • By the Method of Transposition:

  • Simultaneous Linear Equations

  • Elimination of one of the variables through substitution

  • Elimination of one of the variables by making coefficients of any one variable equal

  • Consistent Equations having unique solutions

  • Consistent Equations having infinitely many solutions

  • Inconsistent Equations having No solution

  • How to check if equations do not have any solution:

  • Homogenous Equations:

  • Homogenous equations have infinitely many solutions, if ratios of x and y coefficients are equal.

  • Homogenous equations have only one solution, namely x = 0 and y = 0, if the ratios of the x and y coefficients are not equal.

  • Equations Reducible to Simultaneous Liner Equations:

 

Or, if you wish to capture a terse overview of each Equations Formula, then go through each of the following header-links.

You can also click the header-links to take you to the page discussing in detail the specific Equations concept:

1. How to solve a linear equation:

Transposition method of solving a linear equation:

Example 1:

Solve for x:

x + 2 = 3

Solution:

Transpose 2 to RHS, and reverse its sign.

x = 3 – 2 = 1, so x = 1 is the solution.

2. Simultaneous Equations:

Solving two equations x + y = 3 and x – 2y = 1 for the two different variables x and y so that the solution satisfies both the given equations are called simultaneous equations.

  • Methods of solving simultaneous equations:
1. Elimination Method thru substitution:

Solve the equations:

x – y = 2 and x + y = 1.

Solution:

From the first equation, transpose y to the right by inversing its sign:

x = y + 2,

substitute this value of x, i.e. y + 2 in x in the second equation:

y + 2 + y = 1,

so that, 2y = 1 – 2, 2y = -1, y = -1/2

now, substitute y = -1/2 in any of the two equations to solve for x:
(let us substitute in the first equation)

x – (-1/2) = 2, i.e. x + ½ = 2, so that x = 2 – ½ = 3/2

so, the solution, i.e. values of the two variables x and y, for the pair of equations is x = 3/2 and y = -1/2

2. Elimination thru making coefficients equal:

Solve the system of equations:

2x – 3y = 4 and 3x – 4y = 8.

Solution:

Let us eliminate y by making coefficients of y in the two equations equal.

Multiply first equation with 4, the coefficient of y (disregard the minus sign) in the 2nd equation, and the second with 3, the coefficient of y (here too disregard the minus sign) in the 1st equation as below:

4 × (2x – 3y = 4), i.e. 8x – 12y = 16, and

3 × (3x – 4y = 8), i.e. 9x – 12y = 24.

Now subtract the new 2nd equation from the new 1st, and you will see the y-term gone.

[8x – 12y – 16] – [9x – 12y – 24] = 0

i.e. –x + 8 = 0, i.e. –x = - 8 , i.e. x = 8.

Now, substitute x = 8 in any of the two equations, say in the 1st (the original) to solve for y:

2 × 8 – 3y = 4,

i.e. -3y = 4 – 16,

i.e. -3y = -12,

3y = 12, i.e. y = 12/3 = 4.


Top