Equations Reducible To Linear Form: Methods and Solved Examples

When an equation is written in the form of ax + by + c = 0, where a and b are not both zero and a, b, and c are real integers, it is referred to as a linear equation in two variables x and y.

• Solving equations with two variables using methods like substitution and elimination is a lot easier when the equations are in linear form.
• When the equations are not linear, then, it becomes quite challenging to solve them.
• So, if one tries to solve such an equation directly without using any trick, it becomes a lot messier, and may end up getting the wrong answer or may get stuck.
• The trick involved in solving such equations implies the conversion of the equation into linear form using some rearrangements and algebraic methods.

Key Terms: Linear equation, Algebraic expression, Non-linear equation, Real numbers, Straight line, Intersection point, Parallel line, Fractional variable, Multiplication

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General Form of Pair of Linear Equations

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The general form for the pair of linear equations in two variables x and y is given by

a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0

Where

• a₁, a₂, b₁, b₂, c₁, c₂ are real numbers
• a₁² + b₁² ≠ 0 
• a₂² + b₂² ≠ 0.

An example of a pair of linear equations in two variables is

3x = y and -5x + 3y + 2 = 0

Linear equations with two variables are represented graphically as a straight line. Thus, for a pair of linear equations in two variables, there will be two straight lines with the following three options. They are:

• Two lines intersect.
• Two lines are parallel with each other.

• Two lines coincide.

Also Read:

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How To Reduce To Linear Form?

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The method that is mentioned below is the method of reducing non-linear equations to linear equations using certain rearrangements and suitable substitution methods. Here, the very first step is to substitute a fractional variable or a fractional entity into a single variable by substitution.

Suppose we have two nonlinear equations:-

2/x + 3/y = 13 ; 5/x – 4/y = -2

Step 1: Finding out the suitable variable

First, we will find the fractional variables, here the fractional variables are-

1/x and 1/y, In both the equations.

Step 2: Make sure the chosen variables are common in both the equations

The variables chosen above should be such that they are common in both equations. Here both the fractional variables 1/x and 1/y appear in both the given equations.

Note: The very important task is the choosing of variables that are common in both equations. If you do not choose the variables which are common then, you’re going wrong.

Step 3: Substitute the chosen variables with a single suitable variable

We have carefully found the common fractional variables in both equations. Now we will substitute the chosen variables with a suitable single variable as follows:-

 1/x = p and 1/y = q

Step 4: Replace the given equation

The equations can now take the form

2p + 3q = 13 ;5p – 4q = -3

Step 5: Solve by simple substitution method

Now we will solve the obtained equations by substitution method as follows:-

(2p + 3q) × 4 = 13 × 4 (multiplying by 4)

(5p – 4q) × 3= -2 × 3 (multiplying by 3)

Therefore, we get

8p + 12q = 52    ...(i)

15p – 12q = -6    ...(ii)

On adding equation (i) and (ii), we get

23p = 46

⇒ p = 2

On substituting the value of p in the equation

2p + 3q = 13

⇒ 2×2 + 3q = 13

⇒ q = 3

Hence, the solution of the given pair of equations is:

x = 1/2 and y = 1/3

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Solved Examples

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Ques. Reduce the following equation into the linear form.

1/(x - 2) + 1/y = 3

Ans. The given equation is 1/(x - 2) + 1/y = 3

Let 1/(x – 2) = p, and

1/y = q, then

the equation becomes

p + q = 3

Ques. Solve the given pair of equations by reducing them

1/2x + 1/y = 2;
1/x + 1/2y = 10

Ans. The given equations are

1/2x + 1/y = 2;
1/x + 1/2y = 10

Step 1: Choose the common variable

1/x and 1/y

Step 2: Substitute the chosen variables by a single suitable variable

Let 1/x = p and 1/y = q

Then the equations become

p/2 + q = 2   ...(i)

p + q/2 = 10   ...(ii)

Step 3: Multiply both sides to give a linear equation

Multiplying equation (i)  and (ii) by 2, we get

p + 2q = 4   ...(iii)

2p + q = 20   ...(iv)

On adding equations (iii) and (iv) we get

p + q = 8

On subtracting equations (iii) and (iv) we get

p - q = 16

On adding both the equations, we get

p = 12

On substituting the value of p in equation p + q = 8, we get

q = -4

Now substitute the values

1/x = 12 and 1/y = -4

⇒ x = 1/12; y = -1/4 

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Things to Remember

• The general form for the pair of linear equations in two variables x and y is given by a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0.
• Linear equations with two variables are represented graphically as a straight line.
• While choosing the common variables, make sure that they are really common in both equations.
• Substitute the chosen common variables with a single-term variable and change the given equation accordingly.
• After substituting the equations with the chosen variables, solve the equations by using a simple substitution method.
• After solving the equation, replace the substituted value with the original value and find the answer accordingly.
• Be careful while substituting the variables in the answer and then calculate the value of the exact unknown variable (which is in the question to find out) accordingly.