Wondering why linear algebra is so difficult subject for students? Well, it’s because they are not taught in the right way. We have some excellent methods on how to write and solve linear equations. In this section will help you how to effectively solve step by step problems on linear equations. At the end of the section, student's grasps to make concepts more engaging and relevant. We have also prepared notes based on CBSE latest syllabus for class 7, class 8, class 9 and class 10 students on linear equations separately. Students can use this aids in learning or simply to solve homework tasks and excel their skills. Along with this, NCERT linear equations solutions serve as an extension in classroom lecture and facilitates homework for the students to study at home and allow the teachers to monitor their progress. Here will learn about how to write and solve linear equations along with some solved examples.

Writing Linear Equations in Standard Form

Linear equation is any equation involving one or two variables whose exponents are one.

For $x_1,x_2, x_3,.....,x_n$ are the unknowns and $b_1, b_2,.... ,b_m $ are the constant terms and $a_{11}, a_{12},.......,a_{mn}$ are the coefficients of the system. A general system of m linear equations with n unknowns can be written as:

$a_{11}x_1+a_{12}x_2+....+a_{1n}x_n=b_1$
$a_{21}x_1+a_{22}x_2+....+a_{2n}x_n=b_2$
.
.
.
$a_{m1}x_1+a_{m2}x_2+....+a_{mn}x_n=b_m$

Let us see how to derive standard form of linear equation with one variable, two variable and three variables from the above system.
For variables x and y; a, b, c, s are real numbers and a and b both not equal to 0.
Linear equation with one variable: ax + s = 0
Linear equation with two variables:  ax + by + s = 0
Linear equation with three variables:  ax + by + cz + s = 0

Slope Intercept Form

The slope intercept form of the linear equation of two variables is,
y = mx + c, where 'm' is the slope and 'c' the y-intercept of the line in the graph.

Writing slope-intercept form from the standard form i.e. ax + by = c

by = -ax + c
                                            
$\frac{by}{b}$ = - $\frac{ax}{b}$  +  $\frac{c}{b}$ [ dividing both sides by b]
y = - $\frac{a}{b} x$ + $\frac{c}{b} $, which is the slope-intercept form.

Comparing this with the slope intercept form, y = mx + c,
we get the slope, m = - $ \frac{a}{b}$ and
 the y-intercept is c = $\frac{c}{b}$

Writing Linear Equations from Graphs

To write the linear equations from the graph, we should remember the intercept form of the linear equation.
If  a and b are the x and y - intercepts of a graph of a line, then the equation of a line is given by,
                                                                 $\frac{x}{a}$ + $ \frac{y}{b} $ = 1
For example, if a line meets the x-axis at x = 4 and y-axis at y = -3, then the equation of the line is $\frac{x}{4}$  + $ \frac{y}{-3}$ = 1

Hence the above equation can be rewritten as, $\frac{x}{4}$ - $ \frac{y}{3}$ = 1
Simplify to get standard form:
Multiplying both sides by the LCM of the denominations = 12,
12 ( $\frac{x}{4}$) - 12 $(\frac{y}{3})$ = 12(1)
=>     3x  - 4y = 12, which is the standard form of the equation.
Equation on coordinate looks like: (4, 0) and (-3,0) any points on the graph.

Writing Linear Equation in One Variable Graph

Examples

Example 1: Write the standard form of the linear equation, 2x + 3y + 5 = 17. Hence find the x and y intercepts.
Solution: We have 2x + 3y + 5 = 17 
General standard form, ax + by = c,
To write given equation in standard form, eliminate 5 from the LHS.
2x + 3y + 5 - 5 = 17 - 5  (subtract 5 from both the sides)
=> 2x + 3y = 12
To find the x-intercept : The x-intercept is the point where the line meets the x-axis. (i.e) on the x-axis or y = 0Therefore substituting y = 0 in the above standard form, we get
2x + 3 (0) = 12
=> 2x  + 0 = 12
=>  2x = 12
 =>  x = 12/2 = 6
Therefore the co-ordinate of x-intercept is ( 6,0)
To find the y-intercept : The y-intercept is the point where the line meets the y-axis. (i.e) on the the y-axis or x = 0. Therefore substituting x = 0, in the above standard form,
2 (0) + 3y = 12
=> 0 + 3y = 12
=> 3y = 12
=> y = 12/3 = 4
Therefore the co-ordinate of y-intercept is ( 0,4)
Example 2: Express the equation 3x + 5y = 15 in slope intercept form.  Also find the x and y-intercepts of the equation.
Solution:We have 3x + 5y = 15
5y = - 3x + 15
y = - $\frac{3}{5}$x + $\frac{15}{5}$
y = -  $\frac{3}{5}$x + 3 , which is of the form y = mx + c, the slope - intercept form.

Comparing we get, the slope m = $\frac{-3}{5}$ , and  the y-intercept c = 3

To find the x- intercept: We have 3x + 5y = 15
Substituting y = 0, we get 3x + 5 ( 0) = 15
=> 3x  + 0 = 15
=>        3x = 15
=>          x = 15/3 = 5
The co-ordinate of the x-intercept is ( 5,0)
To find the y-intercept : Substituting x = 0 in 3x + 5y = 15, we get,
3 (0 ) + 5y = 15
=> 0 + 5y = 15
=> 5y = 15
=>  y = 15/5 = 3
The co-ordinate of the y-intercept is ( 0, 3)

Word Problems

While solving linear equations using word problems we should keep in mind the following points.
Step 1: Read the problem carefully and identify the unknown quantities. say x and y.
Step 2: List the given quantities.
Step 3: Convert instructions into equations.
Step 4: Solve the two simultaneous equations for the variables x and y.
Note: Elimination method or cross multiplication method is the most suitable methods.
Example : Ten years ago, father was twelve times as old as his son and ten years hence, he will be twice as old as his son will be. Find their present ages.
Solution:
Let us assume the present age of father as x years and that of the son's y years.Let us table the given data.

  Father's age 
  Son's Age   Relation (or) Equation
   10 years ago     x - 10      y - 10
 Father's age = 12 (Son's age)
            x - 10 = 12( y - 10 )
  Present age 
      x
       y
 
 10 years hence    x + 10   
   y + 10
 Father's age = 2(Son's age)
         x + 10 = 2 ( y + 10)

Hence from the above table, we have the two equations,
x - 10 = 12 ( y - 10)
x + 10 = 2 ( y + 10)
Let us simplify and write the equations in standard form,
x - 10 = 12 y - 120
=> x - 12y = -120 + 10 [ by transposition ]
=>  x - 12y   = -110     --------(1)
Now, x + 10 = 2 ( y + 10 )
x + 10 = 2y + 20
x - 2y  = 20 - 10
x - 2y   = 10                -------(2)

 Subtracting (2) from (1), we get                                                                        
10 y =  120
=>  y = 120/10 = 12
Substituting y = 12 in Equation (1), we get,
x - 12 y = -110
=>  x - 12 ( 12) = -110
=> x - 144    = -110
=> x = 144 - 110 = 34
or  x = 34
Therefore, Father's Present age = 34 years and Son's Present age = 12 years.

Practice Questions

Question 1. Express the equation, 3y + 8x -  24 = 0 in standard form. Hence find the x and y intercepts. Also express the equation in intercept form.

 Question 2. Points A and B are 90 km apart from each other on a highway. A car starts from A and another from B at the same time. If they go in the same direction they meet in 9 hrs and if they go in the opposite direction they  meet in 9/7 hrs. Find their speeds.

Question 3. Study the given graph and write the equation in
a. Intercept form.
b. slope-intercept form
c. standard form
Linear Equation in One Variable Graph