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Linear equation

Linear equation

ax + b = k

Transposition

ax + b = k
ax = k − b
x = (k − b)/a

Linear Equation in one variable


Linear Equations are the starting point of two branches of mathematics. One deals with roots and other deals with solutions. Linear equation can be single variable or multi variable. The complexity of the solutions increases as we move to higher number of variables. As we will see every degree equation can be represented as multi variable linear equations. The point here is that the equations are related and a quadratic and cubic has 3 and 4 unknowns in it. Let us talk about the single variable linear equation. A single variable linear equation is very simple and is taught from the very basic grades or classes. A single variable linear equation looks like this:

ax + b = 0.

There can be many variant of it. ax + b = c or ax + c = bx + d or ax + c = bx, etc.

All we need to know is basic arithmetic. All the laws of basic arithmetic also works in algebra. To solve a linear equation in one variable we perform operations like this:

Let us solve different types of linear equations:

  1. ax + b = 0
    Transpose b to right hand side and divide by a.
    x = -b/a
  2. ax + b = c
    Transpose b to RHS and divide by a.
    x = (c - b)/a
  3. ax + b = cx
    Transpose cx to LHS and b to RHS
    ax - cx = -b
    (a-c)x = -b
    x = -b/(a-c)

Solution


The simplest way to solve a linear equation is to transpose all the constants to RHS and terms containing variable to LHS and divide RHS by the coefficient of resulting LHS.

A single variable linear equation looks like this:

ax + b = 0.

There can be many variant of it. ax + b = c or ax + c = bx + d or ax + c = bx, etc.

All we need to know is basic arithmetic. All the laws of basic arithmetic also works in algebra. To solve a linear equation in one variable we perform operations like this:

Let us solve different types of linear equations:
  1. ax + b = 0
    Transpose b to right hand side and divide by a.
    x = -b/a
  2. ax + b = c
    Transpose b to RHS and divide by a.
    x = (c - b)/a
  3. ax + b = cx
    Transpose cx to LHS and b to RHS
    ax - cx = -b
    (a-c)x = -b
    x = -b/(a-c)
The simplest linear equation is of the form ax + b = 0. Its solution is x = -b/a. It is an equation which contains only one variable of degree 1. Here it is x. We can reduce many equations to this form.

When an equation is reduced then the solution of the equation don't change.

Rules for reducing an equation to this form

  1. The Same number can be added to both sides of the equation.
  2. The Same number can be subtracted from both sides of the equation.
  3. Both sides of the equation can be multiplied by the same non-zero number.
  4. Both sides of the equation can be divided by the same non-zero number.
One or more of these rules can be performed to reduce the equation.



Let us solve the equation 4x + 10 = 2x + 30.

(Solve the equation means find the value of x.)


Solution:

Adding -10 to both sides.
4x + 10 - 10 = 2x + 30 - 10
4x = 2x + 30 - 10
4x = 2x + 20
Subtracting 2x from both sides
4x - 2x = 2x - 2x + 20
4x - 2x = 20
2x = 20
Dividing both sides by 2
2x/2 = 20/2
x = 10


Transposition


As we have seen from the above example that when we add of subtract a quantity from both sides of the equation is same as moving the quantity to the other side with opposite sign. This process of moving is called transposition.


Cross Multiplication


If (ax + b)/(cx + d) = k:l then
(ax + b)l = k(cx + d)
This process is called cross multiplication.

Solution


The simplest way to solve a linear equation is to transpose all the constants to RHS and terms containing variable to LHS and divide RHS by the coefficient of resulting LHS.




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