Solving Linear Equations

An equation is a mathematical statement showing two algebraic or numeric expressions equal. A Linear equation is an equation with one or more variables which do not contain exponents. In other words, a linear equation is formed by adding or subtracting terms which can be numerical multiples or quotients of variables. The name linear is derived from the word ' line' as as the graph of a linear equation in two variable is a straight line in coordinate plane.

Linear equation, an algebraic equation in which each term is either a constant or the product of a constant and single variable.

Solving Linear Equations in One Variable

The solution to a system of equations are the ordered pairs of values of known variables that satisfy all the equations in the system. For a linear equation with one variable, solving the equation involves isolating the variable on one side. This is achieved by using properties of equalities.
  • If a number is added to a variable or to an expression containing the variable, then subtract the same number on either side of the equation.
  • If a number is subtracted to a variable or to an expression containing the variable, then add the same number on either side of the equation.
  • If a variable or an expression containing a variable is multiplied by a number, then divide either side of the equation by the same number.
  • If a variable or an expression containing a variable is divided by a number, then multiply either side of the equation by the same number.

Solved Examples

Question 1: Solve for x, 5x + 9 = 29

Solution:
Given equation 5x + 9 = 29

Step 1:

Subtract 9 from both side

=> 5x + 9 - 9 = 29 - 9

=> 5x = 20

Step 2:

Divide each side by 5,

=> $\frac{5x}{5}$ = $\frac{20}{5}$

=> x = 4.
 

Question 2: Solve 15x + 20a + 4 = 13. Find the value of x, at a = 1.

Solution:
Given equation 15x + 20a + 4 = 13

Step 1:
Subtract 4 from both side

=> 15x + 20a + 4 - 4 = 13 - 4

=> 15x + 20a = 9

Step 2:

Solve for x,

=> 15x = 9 - 20a

Divide each side by 15

=> $\frac{15x}{15}$ = $\frac{9 - 20a}{15}$

=> x = $ \frac{9 - 20a}{15}$

Step 3:

Put a = 1

=> x = $ \frac{9 - 20 * 1}{15}$

=> x = $ \frac{9 - 20}{15}$

=> x = $ \frac{- 11}{15}$
 

Forms of Linear Equation

Forms of linear equation are as follow:

1.  General form: An equation of the form ax + by + c = 0. 
Example, 3x - 5y + 7 = 0 is a general form of equation

where a = 3, b = -5 and C = 7

2.  Standard form: An equation of the form ax + by = c.
Example :  x - 2y = 10 is a standard form of equation

where a = 1, b = -2 and C = 10

3.  Slope Intercept form : An equation of the form y = mx + b.
Example: y = 3x + 5.

where m = 3 and b = 5

System of Linear Equations

A system of linear equation has two or more linear equations involving two or more variables. A system of linear equations is also called 'Simultaneous equations' because any solution should satisfy all the equations simultaneously. A linear equation is an equation with one or more variables but the degree of the highest of the  terms of the equation is one. The equations cannot be solved independently.

A system of linear equations is classified into two parts based on their solutions:
  1. Consistent System, A system has a unique or infinitely many solutions.
  2. Inconsistent System,  An system does not have any solution.

Solved Example

Question: For what value of k does the system of equations have no solution.

6x + 2y = 9

3x + ky = - 7

Solution:
Given system of equations

6x + 2y = 9

3x + ky = - 7

Step 1:
We know that

The system is inconsistent, has no solution

if $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$

By comparing the given equations with the general equations of the system we have

a1 = 6, b= 2, c1 = 9

and b2 = 3, b2 = k, c2 = -7

Step 2:

Since system of equations have no solution

=> $\frac{6}{3} = \frac{2}{k} \neq \frac{9}{7}$

Step 3:

Find the value of k

=> $\frac{6}{3} = \frac{2}{k}$

=> 2 = $ \frac{2}{k}$

=> 2k = 2

=> k = 1

Hence for k = 1 the system of equations have no solution.
 

Solving a System of Equations in Two Variables

A system of two equations in two variables can be solved for both the variables. The substitution and elimination methods are algebraic methods formulated using the properties of equalities. Graphically, the two lines representing the two equations are graphed and the point of intersection if exists is the unique solution of the system. If two lines are parallel then the system is inconsistent and does not have a solution. If the lines drawn are coincident, then the system has infinitely many solutions. The general methods applied for solving a linear system of two variables are,
  1. Substitution method
  2. Elimination method
  3. Graphical solution
Even the other methods like Crame'rs rule and matrix method can be used for solving a system of linear equations in two variables, which are dealt with in Higher Grades.

Solved Examples

Question 1: Solving a system of linear equations by the substitution method.

x + y = 3

5x + 2y = 9
Solution:
Given system of equations

x + y = 3       .....................(1)

5x + 2y = 9   .....................(2)

Step 1:
Apply substitution method to solve equations

Equation (1) => y = 3 - x

Substitute equation (1) in equation (2)

=> 5x + 2 (3 - x) = 9

=> 5x + 6 - 2x = 9

=> 3x = 9 - 6

=> 3x = 3

=> x = 1

Step 2:
Put x = 1 in equation (1)

=> 1 + y = 3

=> y = 3 - 1 = 2

=> y = 2

Hence solution of the system is (x, y) = (1, 2).
 

Question 2: Solving a system of linear equations by the elimination method

3x + y = 2

5x + 2y = - 5

Solution:
Given system of equations

3x + y = 2         ........................(1)

5x + 2y = - 5     ........................(2)

Step 1:

To eliminate y, multiply each side of equation (1) by 2

=> 2(3x + y) = 2 * 2

=> 6x + 2y = 4         ................(3)

Now the coefficients of y terms are same of both the equations.

Step 2:
Subtract the equation (2) from equation (3)

=> 6x + 2y - (5x + 2y) = 4  - (- 5)

=> 6x + 2y - 5x - 2y = 4 + 5

=> x = 9

Step 3:
Put x = 9 in equation (1)

=> 3 * 9 + y = 2

=> 27 + y = 2

=> y = 2 - 27 = - 25

=> Solution of the system is (x, y) = (9, - 25).

 

Question 3: Sketch the graph of x + y = 6.

Solution:
Given linear equation, x + y = 6

The graph of x + y = 6 is shown here

Graph of Linear Equation