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Q2(ii):
Write four solutions for each of the following equations:
(ii) $\pi x + y = 9$
Solution :
Initial Setup & Algebraic Formulation
We are tasked with finding four distinct solutions for the following linear equation in two variables:
$\pi x + y = 9$
A linear equation in two variables has infinitely many solutions. A solution is defined as an ordered pair $(x, y)$ that satisfies the equation. To systematically generate these solutions, we will express one variable in terms of the other.
Step 1: Isolating the Dependent Variable
To make the calculation of ordered pairs straightforward, we isolate the dependent variable ($y$) in terms of the independent variable ($x$). [Per the Subtraction Property of Equality, subtracting $\pi x$ from both sides of the equation yields:]
$y = 9 - \pi x$
We can now substitute arbitrary real numbers for $x$ to compute the corresponding exact values of $y$.
Step 2: Deriving the First Solution
Let us assign $x = 0$. Substituting this value into our isolated equation:
$y = 9 - \pi(0)$
$y = 9 - 0$
$y = 9$
Thus, the first solution is the ordered pair $(0, 9)$. [This point also represents the y-intercept of the linear graph].
Step 3: Deriving the Second Solution
Let us assign $x = 1$. Substituting this value into the equation:
$y = 9 - \pi(1)$
$y = 9 - \pi$
Thus, the second solution is the ordered pair $(1, 9 - \pi)$.
Step 4: Deriving the Third Solution
Let us assign $x = 2$. Substituting this value into the equation:
$y = 9 - \pi(2)$
$y = 9 - 2\pi$
Thus, the third solution is the ordered pair $(2, 9 - 2\pi)$.
Step 5: Deriving the Fourth Solution
Let us assign $x = -1$. Substituting this value into the equation:
$y = 9 - \pi(-1)$
$y = 9 + \pi$
Thus, the fourth solution is the ordered pair $(-1, 9 + \pi)$.
Step 6: Tabular Summary of Solutions
We can organize the derived solutions into a Cartesian coordinate table. This demonstrates the linear relationship and prepares the data for graphical representation.
| Independent Variable ($x$) | Dependent Variable ($y$) | Ordered Pair $(x, y)$ |
|---|---|---|
| $0$ | $9$ | $(0, 9)$ |
| $1$ | $9 - \pi$ | $(1, 9 - \pi)$ |
| $2$ | $9 - 2\pi$ | $(2, 9 - 2\pi)$ |
| $-1$ | $9 + \pi$ | $(-1, 9 + \pi)$ |
Step 7: Graphical Representation of the Linear Equation
Below is the geometric interpretation of the equation $\pi x + y = 9$ plotted on a Cartesian plane. The line represents the locus of all infinite solutions, with our four specific calculated points highlighted. (Note: For spatial plotting purposes, $\pi$ is approximated as $3.14$).
Final Solution: Four valid solutions for the equation $\pi x + y = 9$ are $(0, 9)$, $(1, 9 - \pi)$, $(2, 9 - 2\pi)$, and $(-1, 9 + \pi)$.
More Questions from Class 9 Mathematics Linear Equations in Two Variables EXERCISE 4.2
- Q1: Which one of the following options is true, and why? $y = 3x + 5$ has
- Q2(i): Write four solutions for each of the following equations: (i) $2x + y = 7$
- Q2(iii): Write four solutions for each of the following equations: (iii) $x = 4y$
- Q3(i): Check which of the following are solutions of the equation $x - 2y = 4$ and which are not: (i) $(0, 2)$
- Q3(ii): Check which of the following are solutions of the equation $x - 2y = 4$ and which are not: (ii) $(2, 0)$
- Q3(iii): Check which of the following are solutions of the equation $x - 2y = 4$ and which are not: (iii) $(4, 0)$
- Q3(iv): Check which of the following are solutions of the equation $x - 2y = 4$ and which are not: (iv) $(\sqrt{2}, 4\sqrt{2})$
- Q3(v): Check which of the following are solutions of the equation $x - 2y = 4$ and which are not: (v) $(1, 1)$
- Q4: Find the value of $k$, if $x = 2$, $y = 1$ is a solution of the equation $2x + 3y = k$.
CBSE Solutions for Class 9 Mathematics Linear Equations in Two Variables
Chapters in CBSE - Class 9 Mathematics
Top Tutors who teach Linear Equations in Two Variables
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