**Comparing Coefficient of Linear Equations in Two Variables and Solving :**

To compare the coefficients of linear equations in two variables, the equations must be in the form.

a_{1}x + b_{1}y + c_{1} = 0

a_{2}x + b_{2}y + c_{2} = 0

The following three cases are possible for any given system of linear equations.

(i) a_{1}/a_{2} ≠ b_{1}/b_{2, }we get a unique solution

(ii) a_{1}/a_{2} = a_{1}/a_{2 } = c_{1}/c_{2}, there are infinitely many solutions.

(iii) a_{1}/a_{2} = a_{1}/a_{2 } ≠ c_{1}/c_{2}, there is no solution

**Example 1 :**

Which of the following pairs of linear equations are consistent/inconsistent? if consistent, obtain the solution graphically

(i) x + y = 5

2 x + 2 y = 10

**Solution :**

x + y - 5 = 0

2 x + 2 y - 10 = 0

From the given equations, let us find the values of a_{1}, a_{2}, b_{1}, b_{2}, c_{1} and c_{2}

a_{1} = 1, b_{1 } = 1, c_{1 }= -5

a_{2} = 2, b_{2 } = 2, c_{2 }= -10

a_{1/}a_{2} = 1/2 -------(1)

b_{1}/b_{2} = 1/2 -------(2)

c_{1}/c_{2 }= -5/(-10) = 1/2 -------(3)

This exactly matches the condition,

a_{1}/a_{2} = b_{1}/b_{2} = c_{1}/c_{2}

Hence, the system of equations will have infinitely many solution.

To draw the graph, let us find x and y intercepts.

x + y - 5 = 0

To find x - intercept : Put y = 0 x - 5 = 0 x = 5 (5, 0) |
To find y - intercept : Put x = 0 y - 5 = 0 y = 5 (0, 5) |

Both equations are representing the same line.

(ii) x - y = 8

3 x - 3 y = 16

**Solution :**

x - y – 8 = 0

3 x - 3 y -16 = 0

From the given equations, let us find the values of a_{1}, a_{2}, b_{1}, b_{2}, c_{1} and c_{2}

a_{1} = 1, b_{1 } = -1, c_{1 }= -8

a_{2} = 3, b_{2 } = -3, c_{2 }= -16

a_{1/}a_{2} = 1/3 -------(1)

b_{1}/b_{2} = (-1)/(-3) = 1/3 -------(2)

c_{1}/c_{2 }= -8/(-16) = 1/2 -------(3)

This exactly matches the condition

a_{1/}a_{2} = b_{1}/b_{2 } ≠ c_{1}/c_{2}

Hence, there is no solution.

(iii) 2 x + y - 6 = 0

4 x - 2 y - 4 = 0

**Solution :**

From the given equations, let us find the values of a_{1}, a_{2}, b_{1}, b_{2}, c_{1} and c_{2}

a_{1} = 2, b_{1 } = 1, c_{1 }= -6

a_{2} = 4, b_{2 } = -2, c_{2 }= -4

a_{1/}a_{2} = 2/4 = 1/2 -------(1)

b_{1}/b_{2} = 1/(-2) = -1/2 -------(2)

c_{1}/c_{2 }= -6/(-4) = 3/2 -------(3)

This exactly matches the condition a_{1}/a_{2} ≠ b_{1}/b_{2}

Hence, it has unique solution.

**Graphing 1 ^{st} equation,**

2 x + y - 6 = 0

y = -2x + 6

x-intercept : Put y = 0 -2x + 6 = 0 -2x = -6 x = 3 (3, 0) |
y-intercept : Put x = 0 y = -2(0) + 6 y = 6 (0, 6) |

**Graphing 2 ^{nd} equation,**

4 x - 2 y - 4 = 0

2y = 4x - 4

y = 2x - 2

x-intercept : Put y = 0 2x - 2 = 0 2x = 2 x = 1 (1, 0) |
y-intercept : Put x = 0 y = 2(0) - 2 y = -2 (0, -2) |

The above lines are intersecting at the point (2, 2). So, the solution is x = 2 and y = 2.

(iv) 2 x - 2 y - 2 = 0

4 x - 4 y - 5 = 0

**Solution :**

_{1}, a_{2}, b_{1}, b_{2}, c_{1} and c_{2}

a_{1} = 2, b_{1 } = -2, c_{1 }= -2

a_{2} = 4, b_{2 } = -4, c_{2 }= -5

a_{1/}a_{2} = 2/4 = 1/2 -------(1)

b_{1}/b_{2} = -2/(-4) = 1/2 -------(2)

c_{1}/c_{2 }= -2/(-5) = 2/5 -------(3)

This exactly matches the condition a_{1}/a_{2 } = b_{1}/b_{2} ≠ c_{1}/c_{2}

This exactly matches the condition

a_{1/}a_{2} = b_{1}/b_{2 } ≠ c_{1}/c_{2}

Hence, there is no solution.

After having gone through the stuff given above, we hope that the students would have understood, comparing coefficient of linear equations in two variables and solving.

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