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Q3(i):
On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$, find out whether the following pair of linear equations are consistent, or inconsistent. (i) 3x + 2y = 5 ; 2x – 3y = 7

Solution :

Given: A pair of linear equations in two variables:

Equation 1: $3x + 2y = 5$

Equation 2: $2x - 3y = 7$

To Find: Determine whether the given pair of linear equations is consistent or inconsistent by comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$.

Theoretical Background:

For a pair of linear equations of the form $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$:

  • If $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$, the lines intersect at a unique point, and the system is consistent.
  • If $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$, the lines are coincident, and the system is consistent (dependent).
  • If $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$, the lines are parallel, and the system is inconsistent.

Step 1: Standardizing the Equations

Rewrite the equations in the standard form $ax + by + c = 0$:

Equation 1: $3x + 2y - 5 = 0$

Equation 2: $2x - 3y - 7 = 0$

Step 2: Identifying Coefficients

Comparing with $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$:

For Equation 1: $a_1 = 3$, $b_1 = 2$, $c_1 = -5$

For Equation 2: $a_2 = 2$, $b_2 = -3$, $c_2 = -7$

Step 3: Calculating the Ratios

Ratio of coefficients of $x$: $\frac{a_1}{a_2} = \frac{3}{2}$

Ratio of coefficients of $y$: $\frac{b_1}{b_2} = \frac{2}{-3} = -\frac{2}{3}$

Ratio of constant terms: $\frac{c_1}{c_2} = \frac{-5}{-7} = \frac{5}{7}$

Step 4: Comparing the Ratios

Observe the ratios $\frac{a_1}{a_2}$ and $\frac{b_1}{b_2}$:

Since $\frac{3}{2} \neq -\frac{2}{3}$, it follows that $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$.

Step 5: Conclusion

[Since the condition $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$ is satisfied, the lines intersect at a single point, implying the system has a unique solution.]

Therefore, the pair of linear equations is consistent.

Final Answer: The pair of linear equations is consistent.


More Questions from Class 10 Mathematics Pair of linear equations in two variable EXERCISE 3.1


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