Linear Equation- QA Formulae

 Important points:

• The values of the variables that make a linear equation true are called the solution of the linear equation.
•  The solution of a linear equation is unaffected if the same number is added to, is subtracted from, is multiplied by, or divided both sides of the equation.

 Equations with two variables:

• Consider two equations, ax + by = c and mx + ny = p. Each of these equations represents two lines on the x–y coordinate plane. The values of x and y that satisfy both of these equations give the point of intersection.
•  If the coefficients satisfy   then the slope of the two equations is equal. So, they are parallel to each other. Hence, there is no point of intersection. Therefore, there is no common solution.

• If the coefficients satisfy  then the slope is different. So, they intersect with each other at a single point. Hence, the equations have a single common solution.

• If the coefficients satisfy  then the two lines are similar and have infinite points in common. So, the equations have infinite common solutions.

 General procedure to solve linear equations:

• Aggregate the constant terms and variable terms.

• For equations with more than one variable, eliminate the variables by substituting them with equations.

• For two equations with two variables, x and y, express y in terms of x, and substitute it in the other equation.

 Example:

Let x + y = 8 and x + 3y = 12

 Then, x = 8 – y (from equation 1) 

By substituting this in equation 2, we get the following: 

8 – y + 3y = 12

Hence, y = 2 and x = 6.

Note:

For equations of the form ‘ax + by = c’ and ‘mx + ny = p’, find the LCM of b and n. Multiply each equation with a constant to make the coefficient of y equal to the LCM. Then, subtract equation 2 from equation 1.

Example: Let 2x + 3y = 13 and 3x + 4y = 18 be equations (1) and (2). 

The LCM of 3 and 4 is 12. 

By multiplying equation (1) by 4 and equation (2) by 3, we get the following: 

8x + 12y = 52 and 9x + 12y = 54.

(2) – (1) gives x = 2 and y = 3. 

The same procedure can be followed for the coefficients of ‘x’ as well to find the answer. 

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