9th Class Mathematics Linear Equations in Two Variables Conditions

       Conditions

For pair of linear equations in two variable is......

\[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0\]

\[{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0\]

(i) If \[\frac{{{a}_{1}}}{{{a}_{2}}}\ne \frac{{{b}_{1}}}{{{b}_{2}}}\] then lines are intersecting.

(ii) If \[\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{{{b}_{1}}}{{{b}_{2}}}=\frac{{{c}_{1}}}{{{c}_{2}}}\] then lines are coincident.

(iii) If\[\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{{{b}_{1}}}{{{b}_{2}}}\ne \frac{{{c}_{1}}}{{{c}_{2}}}\] then lines are parallel.  

Solve for \[\mathbf{x}\] and y; graphically.

\[x-y-1=0\]

\[2x+y-8=0\]  

Solution:

We have

\[x-y-1=0\]                                                     ....(i)

\[\Rightarrow \]\[x=y+1\]

Now locate the points (1, 0), (2, 1) and (3, 2) on the graph paper and draw a line passing through it

\[2x+y-8=0\]                      \[\Rightarrow \]\[y=8-2x\]

Now locate the point (1, 6), (2, 4) and (3, 2) on the same graph and draw another line passes through these points

In the graph the coordinate of point of intersection is (3, 2). Therefore, \[x=3\] and y = 2 is the solution of given pair of linear equations.    

• Do you know that ancient Egyptians Rhind papyrus near about 1650 B.C. had ability to solve linear equation in one unknown. The extension of linear equation was further used in linear algebra in the early 1840 by William Rowan Hamilton, which describe mechanics in three-dimensional space.

•   The general form of linear equation in two variables is \[ax+by+c=0\], where a, \[b\ne 0\]
•   \[x=p\]and \[y=q\] is called the solution of a linear equation \[ax+by+c=0\] if \[ap+bq+c=0\]  

  Which one of the following options is correct?

(a) Linear equation in two variable is in the form of \[\text{a}x+\text{by}+\text{c}=0\], where \[\text{a}=0\]

(b) Linear equation in two variable is in the form \[px+q\text{y}+\text{c}=0\]

(c) Linear equation is in the form of \[sx+t\text{y}+r=0\], where s, r and t are constants and s, \[t\ne 0\]

(d) Linear equation in three variable is \[ax+b\text{y}+c=0\], where a, b, c are constants and a,\[b\ne 0\]

(e) None of these  

Answer: (c)

Explanation:

Compare with the general formula, it is clear that option (c) is correct.  

  Which one of the following statements is false?

(a) The equation of x-axis is y = 0

(b) The equation of y-axis is \[x=0\]

(c) The equation of a line parallel to x-axis at a distance a from origin is \[x=a\]

(d) The equation of a line which is parallel to y-axis at a distance a from origin be \[x=a\]

(e) None of these

Answer: (c)

Explanation:

Option (c) is false because equation of required line will be y = a  

  The area bounded by y-axis, \[2x+3y=12\] and \[x-y=1\] is.... .

(a) 7.5 square unit                          

(b) 13.5 square unit

(c) 4.5 square unit                           

(d) 75 square unit

(e) None of these

Answer: (a)    

  In the following pair of linear equation the value of \[x\] and y is \[2x+3y=-5\] and \[3x-2y=12\]:

(a) (2, 3)                                              

(b) (-2,-3)

(c) (-2, 3)                                             

(d) (2, 3)

(e) None of these

Answer: (d)  

  The coordinate of points where lines \[x+3y-6=0\] and \[2x-3y-12=0\] intersect y-axis:

(a) (0, 2), (-4, 0)                                

(b) (2, 0), (0, -4)

(c) (0, 2), (0, -4)                                

(d) (0, 2), (-4, 0)

(e) None of these  

Answer: (c)