Exercise 4.3: Linear Equations in Two Variables — NCERT Solutions

Find at least two solutions of $x + y = 5$ (using $y = 5 - x$):

| $x$ | $y = 5 - x$ | Point | |-----|------------|-------| | $0$ | $5$ | $(0, 5)$ | | $5$ | $0$ | $(5, 0)$ | | $2$ | $3$ | $(2, 3)$ |

Plotting: Mark the points $(0,5)$, $(5,0)$ and $(2,3)$ on the coordinate plane.

Line: Since all three points lie in a straight line, join them and extend both ways. This straight line is the graph of $x + y = 5$.

The line meets the $y$-axis at $(0,5)$ and the $x$-axis at $(5,0)$.

Rewrite as $y = 2x - 4$ and prepare a table:

| $x$ | $y = 2x - 4$ | Point | |-----|-------------|-------| | $0$ | $-4$ | $(0, -4)$ | | $2$ | $0$ | $(2, 0)$ | | $3$ | $2$ | $(3, 2)$ |

Plot the points $(0,-4)$, $(2,0)$, $(3,2)$ and join them with a straight line.

Intercepts:

• Cuts the $x$-axis where $y = 0$: at $(2, 0)$.
• Cuts the $y$-axis where $x = 0$: at $(0, -4)$.

Prepare a table of solutions for $y = 3x$:

| $x$ | $y = 3x$ | Point | |-----|----------|-------| | $0$ | $0$ | $(0, 0)$ | | $1$ | $3$ | $(1, 3)$ | | $-1$ | $-3$ | $(-1, -3)$ |

Plot the points and join them to get a straight line.

Verification: When $x = 0$, $y = 3(0) = 0$, giving the point $(0,0)$, which is the origin. Since the origin satisfies the equation, the graph passes through the origin.

Every equation of the form $y = mx$ (with $c = 0$) passes through the origin.