1. Graphical Representation
A linear equation in two variables ($ax + by + c = 0$) represents a straight line
on the Cartesian plane. Every point on this line is a solution to the equation, and every solution
to the equation is a point on this line.
2. Steps to Draw the Graph
To draw the graph of a linear equation, follow these steps:
- Express $y$ in terms of $x$ (or $x$ in terms of $y$). e.g., $y = 3x - 2$.
- Choose at least two convenient values for $x$ and find the corresponding values of $y$.
- Prepare a table of these solutions $(x, y)$.
- Plot these points on a graph paper.
- Join the points with a straight line and extend it in both directions.
Example: Graph of $x + y = 4$
- If $x = 0$, then $0 + y = 4 \Rightarrow y = 4$. Point: $(0, 4)$
- If $x = 4$, then $4 + y = 4 \Rightarrow y = 0$. Point: $(4, 0)$
- If $x = 2$, then $2 + y = 4 \Rightarrow y = 2$. Point: $(2, 2)$
Plotting $(0, 4), (4, 0), (2, 2)$ and joining them gives the graph.
3. Equations of Lines Parallel to Axes
- Equation of X-axis: $y = 0$
- Equation of Y-axis: $x = 0$
- Line parallel to X-axis: $y = k$ (where $k$ is a constant).
- Line parallel to Y-axis: $x = k$ (where $k$ is a constant).
Question 1
The graph of the linear equation $2x + 3y = 6$ cuts the $y$-axis at
the point:
A. $(2, 0)$
B. $(0, 2)$
C. $(3, 0)$
D. $(0, 3)$
Correct Answer: B
On the $y$-axis, $x = 0$. Using $x=0$ in the equation: $2(0) + 3y = 6 \Rightarrow 3y = 6
\Rightarrow y = 2$. So the point is $(0, 2)$.
Question 2
The equation $x = 7$, in two variables, can be written as:
A. $1 \cdot x + 0 \cdot y = 7$
B. $1 \cdot x + 1 \cdot y = 7$
C. $0 \cdot x + 1 \cdot y = 7$
D. $0 \cdot x + 0 \cdot y = 7$
Correct Answer: A
Since $y$ is missing, its coefficient is 0. Thus, $1 \cdot x + 0 \cdot y = 7$.
Question 3
Any point on the line $y = x$ is of the form:
A. $(a, a)$
B. $(0, a)$
C. $(a, 0)$
D. $(a, -a)$
Correct Answer: A
Since $y = x$, the ordinate must equal the abscissa. Thus, points are like $(1, 1), (2, 2), (-5,
-5)$, generally $(a, a)$.
Question 4
The graph of $y = 6$ is a line:
A. Parallel to x-axis at a distance of 6 units from
the origin
B. Parallel to y-axis at a distance of 6 units from the origin
C. Making an intercept 6 on the x-axis
D. Making an intercept 6 on both axes
Correct Answer: A
$y = k$ represents a line parallel to the x-axis. Here $y = 6$ is a horizontal line 6 units
above the x-axis.
Question 5
Which of the following points lies on the line $y = 2x + 3$?
A. $(2, 8)$
B. $(3, 9)$
C. $(4, 12)$
D. $(5, 15)$
Correct Answer: B
Substitute $x=3$: $y = 2(3) + 3 = 6 + 3 = 9$. Since LHS = RHS, $(3, 9)$ lies on the line.
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