NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.1 Question 4(ii)
NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables
Question:
Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically:
\( x - y = 8 \),
\( 3x - 3y = 16 \)
Given:
The given pair of linear equations is:
1) \( x - y = 8 \)
2) \( 3x - 3y = 16 \)
To Find:
- Whether the given pair of equations is consistent or inconsistent.
- If consistent, obtain the solution graphically.
Formula:
- Two linear equations in two variables are consistent if they have at least one solution (i.e., they intersect or are coincident).
- For equations in the form \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \):
-
Consistent if
\( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \), or
\( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \) (Inconsistent if all three ratios are equal). - To solve graphically, plot both equations and find their point of intersection.
Solution:
Step 1: Write both equations in standard form.
$$ \begin{align*} \text{Equation 1:} \quad & x - y = 8 \\ \text{Equation 2:} \quad & 3x - 3y = 16 \\ \end{align*} $$Step 2: Express both equations in the form \( a_1x + b_1y + c_1 = 0 \).
$$ \begin{align*} x - y - 8 &= 0 \\ 3x - 3y - 16 &= 0 \\ \end{align*} $$Step 3: Find the ratios \( \frac{a_1}{a_2} \), \( \frac{b_1}{b_2} \), and \( \frac{c_1}{c_2} \).
Here, \( a_1 = 1, \ b_1 = -1, \ c_1 = -8 \)
\( a_2 = 3, \ b_2 = -3, \ c_2 = -16 \)
Step 4: Analyze the ratios to determine consistency.
Since \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \),
the pair of equations is inconsistent.
Step 5: State the conclusion.
The equations represent parallel lines (same slope, different intercepts).
Therefore, they do not intersect and have no solution.
Result:
The given pair of linear equations is inconsistent.
There is no solution.
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