Q2(ii):

Express the following linear equations in the form $ax + by + c = 0$ and indicate the values of $a$, $b$ and $c$ in each case: (ii) $x - \frac{y}{5} - 10 = 0$

Initial Setup & Theoretical Foundation

The standard form of a linear equation in two variables is defined universally as:

$ax + by + c = 0$

Where:

• $x$ and $y$ are the variables.
• $a$, $b$, and $c$ are real numbers ($\mathbb{R}$).
• $a$ and $b$ are not both zero (i.e., $a^2 + b^2 \neq 0$).

We are given the linear equation:

$x - \frac{y}{5} - 10 = 0$

Step 1: Structural Alignment of the Equation

To express the given equation strictly in the form $ax + by + c = 0$, we must isolate the numerical coefficients for each term. We rewrite the equation by explicitly stating the multiplier for each variable and converting subtractions into the addition of negative quantities [Per the definition of algebraic subtraction: $A - B = A + (-B)$].

• The term $x$ can be written with an explicit coefficient of $1$: $1 \cdot x$
• The term $-\frac{y}{5}$ can be factored into a fractional coefficient multiplied by $y$: $\left(-\frac{1}{5}\right)y$
• The constant term $-10$ can be written as an added negative integer: $+ (-10)$

Substituting these explicit forms back into the equation yields:

$1x + \left(-\frac{1}{5}\right)y + (-10) = 0$

Step 2: Extraction of Coefficients

By directly comparing our structurally aligned equation to the standard form $ax + by + c = 0$, we establish a one-to-one correspondence between the coefficients:

Step 3: Alternative Integer Representation (Rigorous Extension)

While the fractional coefficients extracted above are perfectly correct, linear equations are frequently scaled to clear denominators for computational simplicity. By applying the Multiplication Property of Equality, we can multiply the entire equation by $5$ (the Least Common Multiple of the denominators):

$5 \cdot \left(x - \frac{y}{5} - 10\right) = 5 \cdot (0)$

$5x - y - 50 = 0$

Rewriting this in explicit standard form gives $5x + (-1)y + (-50) = 0$. In this equivalent integer-based representation, the coefficients would be $a = 5$, $b = -1$, and $c = -50$. Both sets of coefficients define the exact same geometric line.

Graphical Analysis of the Linear Equation

To visualize the geometric truth of this equation, we plot the line $x - \frac{y}{5} - 10 = 0$. Solving for $y$ yields the slope-intercept form $y = 5x - 50$. The line intersects the Cartesian axes at $(10, 0)$ and $(0, -50)$.

Final Solution: The equation expressed in the standard form $ax + by + c = 0$ is $1x + \left(-\frac{1}{5}\right)y + (-10) = 0$. The corresponding values for the coefficients are $a = 1$, $b = -\frac{1}{5}$, and $c = -10$.