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Q2(iii):
Write four solutions for each of the following equations:
(iii) $x = 4y$
Solution :
Initial Setup & Theoretical Foundation
We are given the linear equation in two variables:
$x = 4y$
[Per the fundamental theorem of linear algebra], a linear equation in two variables of the form $ax + by + c = 0$ has infinitely many solutions. This is because for every arbitrary real value assigned to the independent variable, there exists a unique corresponding real value for the dependent variable. To find four distinct solutions, we will systematically assign four different real values to $y$ and solve for the corresponding values of $x$.
Step 1: Deriving the First Solution
Let us assign the value $y = 0$. Substituting this into the given equation:
$x = 4(0)$
$x = 0$
Thus, the first ordered pair $(x, y)$ that satisfies the equation is $(0, 0)$.
Step 2: Deriving the Second Solution
Let us assign the value $y = 1$. Substituting this into the given equation:
$x = 4(1)$
$x = 4$
Thus, the second ordered pair $(x, y)$ that satisfies the equation is $(4, 1)$.
Step 3: Deriving the Third Solution
Let us assign the value $y = 2$. Substituting this into the given equation:
$x = 4(2)$
$x = 8$
Thus, the third ordered pair $(x, y)$ that satisfies the equation is $(8, 2)$.
Step 4: Deriving the Fourth Solution
Let us assign a negative integer to demonstrate solutions in the third quadrant. Let $y = -1$. Substituting this into the given equation:
$x = 4(-1)$
$x = -4$
Thus, the fourth ordered pair $(x, y)$ that satisfies the equation is $(-4, -1)$.
Tabular Representation of Solutions
We can organize the derived solutions into a coordinate table to clearly map the relationship between $x$ and $y$.
| Solution Number | Value of $y$ | Calculated Value of $x$ ($x = 4y$) | Ordered Pair $(x, y)$ |
|---|---|---|---|
| 1 | $0$ | $0$ | $(0, 0)$ |
| 2 | $1$ | $4$ | $(4, 1)$ |
| 3 | $2$ | $8$ | $(8, 2)$ |
| 4 | $-1$ | $-4$ | $(-4, -1)$ |
Graphical Verification
Plotting these points on a Cartesian plane confirms that they are collinear, forming the straight line defined by $x = 4y$.
Final Solution: Four valid solutions for the equation $x = 4y$ are $(0, 0)$, $(4, 1)$, $(8, 2)$, and $(-4, -1)$.
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(ii): Write four solutions for each of the following equations: (ii) $\pi x + y = 9$
- 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
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