Initial Setup & Theoretical Foundation

We are given the linear equation:

$2x = -5y$

The objective is to express this equation in the general (standard) form of a linear equation in two variables, which is defined as:

$ax + by + c = 0$

where $a$, $b$, and $c$ are real numbers, and $a$ and $b$ are not both zero simultaneously (i.e., $a^2 + b^2 \neq 0$).

Step 1: Algebraic Transposition

To convert the given equation into the standard form, all non-zero terms must be collected on the left-hand side (LHS) of the equation, leaving zero on the right-hand side (RHS).

We apply the Addition Property of Equality [which states that adding the same quantity to both sides of an equation preserves the equality]. We add $5y$ to both sides of the equation:

$2x + 5y = -5y + 5y$

$2x + 5y = 0$

Step 2: Structuring the Standard Form

The standard form requires three distinct terms: an $x$-term, a $y$-term, and a constant term $c$. In our current equation, $2x + 5y = 0$, the constant term is absent. Mathematically, an absent term implies a coefficient or value of zero. We explicitly introduce zero as the constant term to perfectly match the $ax + by + c = 0$ template:

$2x + 5y + 0 = 0$

Step 3: Coefficient Extraction

By directly comparing our structured equation with the general form $ax + by + c = 0$, we can extract the corresponding real number coefficients:

• The coefficient of $x$ corresponds to $a$. Thus, $a = 2$.
• The coefficient of $y$ corresponds to $b$. Thus, $b = 5$.
• The constant term corresponds to $c$. Thus, $c = 0$.

Graphical Representation (Geometric Interpretation)

A linear equation in two variables geometrically represents a straight line on a Cartesian plane. Because the constant term $c = 0$, the line must pass through the origin $(0,0)$. We can verify this by finding two points that satisfy $2x + 5y = 0$:

• If $x = 0$, then $5y = 0 \implies y = 0$. Point: $(0, 0)$
• If $x = 5$, then $10 + 5y = 0 \implies 5y = -10 \implies y = -2$. Point: $(5, -2)$
• If $x = -5$, then $-10 + 5y = 0 \implies 5y = 10 \implies y = 2$. Point: $(-5, 2)$

Final Solution: The linear equation expressed in standard form is $2x + 5y + 0 = 0$. The corresponding coefficient values are $a = 2$, $b = 5$, and $c = 0$.

Initial Setup & Given Equation

We are given the linear equation:

$3x + 2 = 0$

The objective is to express this equation in the standard two-variable form, $ax + by + c = 0$, and to identify the corresponding real coefficients $a$, $b$, and $c$.

Step 1: Algebraic Transformation to Standard Form

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

$ax + by + c = 0$

where $x$ and $y$ are variables, and $a$, $b$, and $c$ are real numbers such that $a$ and $b$ are not simultaneously zero [Per the fundamental definition of a linear equation in two variables].

Observing the given equation $3x + 2 = 0$, we note the absence of the $y$-variable. To represent this equation in a two-variable format without altering its mathematical equivalence, we introduce the variable $y$ with a coefficient of $0$. [Since $0 \cdot y = 0$ for any real number $y$, the additive identity property ensures the equation's value remains unchanged].

Rewriting the equation yields:

$3x + 0 \cdot y + 2 = 0$

Step 2: Extraction of Coefficients

By aligning our transformed equation directly with the standard form, we can map the corresponding coefficients:

• Standard Form: $ax + by + c = 0$
• Transformed Equation: $3x + 0y + 2 = 0$

Comparing the terms positionally:

• The coefficient of $x$ is $a = 3$.
• The coefficient of $y$ is $b = 0$.
• The constant term is $c = 2$.

Geometric Interpretation (Graphical Representation)

Geometrically, solving the equation $3x + 2 = 0$ for $x$ yields $x = -\frac{2}{3}$. In a two-dimensional Cartesian coordinate system, an equation of the form $x = k$ (where $k$ is a constant) represents a vertical line parallel to the $y$-axis. The line intersects the $x$-axis at the coordinate $(-\frac{2}{3}, 0)$. Because $y$ can take any real value while $x$ remains constant, the coefficient of $y$ is effectively zero.

Final Conclusion

Final Solution: The equation expressed in the standard form $ax + by + c = 0$ is $3x + 0y + 2 = 0$. The corresponding coefficients are $a = 3$, $b = 0$, and $c = 2$.

Initial Setup & Theoretical Foundation

We are given the linear equation:

$y - 2 = 0$

The objective is to express this equation in the standard general form of a linear equation in two variables, which is defined as:

$ax + by + c = 0$

[Per the fundamental theorem of linear algebra in two dimensions, $x$ and $y$ are the variables, while $a$, $b$, and $c$ are real numbers known as coefficients and constants, respectively. For the equation to represent a line in a two-dimensional Cartesian plane, $a$ and $b$ cannot both be simultaneously zero ($a^2 + b^2 \neq 0$)].

Step 1: Algebraic Manipulation to Standard Form

Observing the given equation $y - 2 = 0$, we note that the variable $x$ is absent. To map this to the two-variable standard form $ax + by + c = 0$, we must introduce the variable $x$ without altering the mathematical truth or value of the original equation.

[By the Zero Property of Multiplication, any real number multiplied by zero equals zero. Therefore, we can introduce the term $0 \cdot x$].

Rewriting the equation with the $x$ term:

$0 \cdot x + y - 2 = 0$

To make the coefficients explicitly clear, we can write the implied coefficient of $y$ (which is $1$) and express the subtraction of $2$ as the addition of a negative constant:

$0x + 1y + (-2) = 0$

Step 2: Extraction of Coefficients

Now, we perform a direct term-by-term comparison between our expanded equation and the standard form:

• Standard Form: $ax + by + c = 0$
• Expanded Equation: $0x + 1y + (-2) = 0$

By equating the corresponding coefficients, we derive:

• The coefficient of $x$ is $a \implies a = 0$
• The coefficient of $y$ is $b \implies b = 1$
• The constant term is $c \implies c = -2$

Geometric Interpretation (Visual Aid)

Geometrically, a linear equation where $a = 0$ and $b \neq 0$ represents a horizontal line parallel to the x-axis. The equation $0x + 1y - 2 = 0$ simplifies to $y = 2$. This means that for any real value of $x$, the value of $y$ remains strictly $2$.

Final Solution: The equation expressed in the standard form $ax + by + c = 0$ is $0x + 1y - 2 = 0$. The corresponding values of the coefficients are $a = 0$, $b = 1$, and $c = -2$.

Given Equation & Initial Setup

We are given the following linear equation in one variable:

$5 = 2x$

Our objective is to express this equation in the general standard form of a linear equation in two variables, which is given by:

$ax + by + c = 0$

where $a$, $b$, and $c$ are real numbers, and $a$ and $b$ are not both zero ($a^2 + b^2 \neq 0$).

Step 1: Algebraic Manipulation to Standard Form

To convert the given equation into the standard form, we must transpose all terms to one side of the equality so that the other side becomes zero. [Per the properties of equality, subtracting the same value from both sides preserves the truth value of the equation].

Subtracting $5$ from both sides of the equation:

$5 - 5 = 2x - 5$

$0 = 2x - 5$

By the symmetric property of equality ($A = B \implies B = A$), we can rewrite this as:

$2x - 5 = 0$

Note: Alternatively, subtracting $2x$ from both sides yields $-2x + 5 = 0$. Both forms are mathematically valid, but maintaining a positive leading coefficient is standard convention.

Step 2: Introducing the Second Variable

The standard form requires two variables, $x$ and $y$. Since the variable $y$ is absent from our current equation, it implies that the coefficient of $y$ is exactly zero. [Any real number multiplied by zero is zero, hence $0 \cdot y = 0$].

We explicitly introduce the $y$ term to match the $ax + by + c = 0$ structure:

$2x + 0y - 5 = 0$

Step 3: Identification of Coefficients

Now, we align our derived equation with the general standard form to extract the values of the constants $a$, $b$, and $c$:

• Standard Form: $ax + by + c = 0$
• Derived Equation: $2x + 0y + (-5) = 0$

By direct comparison of the corresponding terms:

• The coefficient of $x$ is $a \implies a = 2$
• The coefficient of $y$ is $b \implies b = 0$
• The constant term is $c \implies c = -5$

Geometric Representation (Graphical Analysis)

A linear equation in two variables where one coefficient is zero represents a line parallel to one of the coordinate axes. The equation $2x - 5 = 0$ simplifies to $x = 2.5$. This represents a vertical line parallel to the Y-axis, intersecting the X-axis at exactly $(2.5, 0)$.

Final Solution: The equation expressed in the standard form $ax + by + c = 0$ is $2x + 0y - 5 = 0$. The corresponding values of the coefficients are $a = 2$, $b = 0$, and $c = -5$.

(Note: The alternative valid form $-2x + 0y + 5 = 0$ yields $a = -2$, $b = 0$, $c = 5$.)

Initial Setup & Theoretical Foundation

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

$ax + by + c = 0$

where $a$, $b$, and $c$ are real numbers ($a, b, c \in \mathbb{R}$), and the coefficients $a$ and $b$ are not simultaneously zero (often denoted as $a^2 + b^2 \neq 0$).

We are given the following linear equation:

$2x + 3y = 9.3\overline{5}$

Step 1: Transposition to Standard Form

To map the given equation to the standard form, we must collect all terms on the left-hand side of the equality, leaving exactly zero on the right-hand side.

[Per the Subtraction Property of Equality], we subtract the constant term $9.3\overline{5}$ from both sides of the equation. Note that $9.3\overline{5}$ is a rational real number (a non-terminating, repeating decimal), and it is treated algebraically as a single constant value.

$2x + 3y - 9.3\overline{5} = 9.3\overline{5} - 9.3\overline{5}$

$2x + 3y - 9.3\overline{5} = 0$

Step 2: Coefficient Extraction and Comparative Analysis

To strictly match the $ax + by + c = 0$ structure, we can conceptually express the subtraction of the constant as the addition of a negative constant:

$2x + 3y + (-9.3\overline{5}) = 0$

By applying the principle of polynomial identity, we equate the corresponding coefficients from our derived equation directly to the standard form variables:

• The coefficient of the $x$-term corresponds to $a \implies a = 2$
• The coefficient of the $y$-term corresponds to $b \implies b = 3$
• The constant term corresponds to $c \implies c = -9.3\overline{5}$

Final Solution: The linear equation expressed in standard form is $2x + 3y - 9.3\overline{5} = 0$. The corresponding coefficient values are $a = 2$, $b = 3$, and $c = -9.3\overline{5}$.

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 $c = -10$.

Given Variables & Initial Setup

We are given the following linear equation in two variables:

$-2x + 3y = 6$

The objective is to express this equation in the general (standard) form of a linear equation in two variables, which is defined as:

$ax + by + c = 0$

where $a$, $b$, and $c$ are real numbers, and $a$ and $b$ are not both zero (i.e., $a^2 + b^2 \neq 0$).

Step 1: Algebraic Manipulation to Standard Form

To convert the given equation into the standard form, we must ensure that all terms (variables and constants) are collected on the left-hand side (LHS) of the equation, leaving exactly zero on the right-hand side (RHS).

• Original Equation: $-2x + 3y = 6$
• Transformation: Apply the Subtraction Property of Equality [Subtracting $6$ from both sides of the equation to nullify the RHS].
• $-2x + 3y - 6 = 6 - 6$
• $-2x + 3y - 6 = 0$

Step 2: Identification of Coefficients

Now, we align our transformed equation with the standard theoretical model to extract the corresponding coefficients.

Standard Form: $ax + by + c = 0$

Derived Equation: $(-2)x + (3)y + (-6) = 0$

By direct comparison of the corresponding terms, we identify the values:

• The coefficient of $x$, denoted as $a$, is $-2$.
• The coefficient of $y$, denoted as $b$, is $3$.
• The constant term, denoted as $c$, is $-6$.

Note on Mathematical Equivalence: Multiplying the entire equation by $-1$ yields $2x - 3y + 6 = 0$. In this equivalent standard form, the coefficients would be $a = 2$, $b = -3$, and $c = 6$. Both representations are mathematically rigorous and correct, though the direct transposition (yielding $a = -2$) is the primary expected procedure.

Graphical Representation of the Linear Equation

To provide a complete analytical perspective, the equation $-2x + 3y = 6$ represents a straight line on a Cartesian plane. The intercepts are calculated as follows:

• x-intercept: Set $y = 0 \implies -2x = 6 \implies x = -3$. Coordinate: $(-3, 0)$
• y-intercept: Set $x = 0 \implies 3y = 6 \implies y = 2$. Coordinate: $(0, 2)$

Final Solution: The linear equation expressed in the standard form $ax + by + c = 0$ is $-2x + 3y - 6 = 0$. The corresponding values of the coefficients are $a = -2$, $b = 3$, and $c = -6$.

Initial Setup & Objective

We are given the following linear equation in two variables:

$x = 3y$

The objective is to express this equation in the general standard form of a linear equation in two variables, which is given by:

$ax + by + c = 0$

where $a$, $b$, and $c$ are real numbers, and $a$ and $b$ are not both zero ($a^2 + b^2 \neq 0$). After transforming the equation, we must identify the specific numerical values for the coefficients $a$, $b$, and the constant term $c$.

Step 1: Algebraic Manipulation to Standard Form

To convert the given equation into the standard form, all non-zero terms must be transposed to one side of the equality, leaving a zero on the opposite side.

Given:

$x = 3y$

We subtract $3y$ from both sides of the equation [Per the Subtraction Property of Equality, which states that if $A = B$, then $A - C = B - C$]:

$x - 3y = 3y - 3y$

$x - 3y = 0$

Step 2: Explicit Representation of Coefficients

The standard form requires explicit representation of the $x$-term, the $y$-term, and the constant term $c$. We can rewrite the expression $x - 3y = 0$ by explicitly stating the hidden multiplicative identity (1) and the additive identity (0) [By the definition of the standard form of a linear equation]:

$1 \cdot x + (-3) \cdot y + 0 = 0$

Step 3: Identification of Parameters

By directly comparing our expanded equation to the standard form $ax + by + c = 0$, we can map the corresponding coefficients:

• The coefficient of $x$ is $a$. Therefore, $a = 1$.
• The coefficient of $y$ is $b$. Therefore, $b = -3$.
• The constant term is $c$. Since there is no constant term present in the original equation, $c = 0$.

Note: An alternative, equally valid standard form can be achieved by subtracting $x$ from both sides (yielding $-x + 3y + 0 = 0$), which would result in $a = -1$, $b = 3$, and $c = 0$. However, it is conventional mathematical practice to maintain a positive leading coefficient for $x$.

Graphical Representation & Analysis

Because the constant term $c = 0$, the linear equation $x = 3y$ (or $y = \frac{1}{3}x$) represents a straight line that passes directly through the origin $(0,0)$. For every 3 units moved along the x-axis, the line moves 1 unit along the y-axis.

Final Solution: The linear equation expressed in the standard form $ax + by + c = 0$ is $1x - 3y + 0 = 0$. The corresponding values for the coefficients are $a = 1$, $b = -3$, and $c = 0$.

Step 1: Variable Assignment & Initial Setup

To construct a mathematical model of the given real-world scenario, we first define the unknown quantities using algebraic variables. The problem explicitly provides the variable assignments:

• Let the cost of a single notebook be represented by the variable $x$.
• Let the cost of a single pen be represented by the variable $y$.

[These variables represent continuous, non-negative rational numbers, as they denote monetary cost: $x, y \ge 0$].

Step 2: Mathematical Translation of the Statement

The core logical condition provided is: "The cost of a notebook is twice the cost of a pen."

We break down this statement into its algebraic equivalents:

• "The cost of a notebook" translates to $x$.
• "is" translates to the equality operator $=$.
• "twice the cost of a pen" translates to the scalar multiplication $2 \times y$, or simply $2y$.

Step 3: Formulation of the Linear Equation

Equating the translated expressions yields the direct algebraic relationship:

$x = 2y$

[This equation demonstrates a direct proportional relationship between the cost of the notebook and the cost of the pen, where the constant of proportionality is $2$].

Step 4: Standardization of the Equation

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

$ax + by + c = 0$

where $a$, $b$, and $c$ are real numbers, and $a$ and $b$ are not both zero ($a^2 + b^2 \neq 0$).

To convert our formulated equation $x = 2y$ into the standard form, we apply the Subtraction Property of Equality by subtracting $2y$ from both sides of the equation:

$x - 2y = 2y - 2y$

$x - 2y = 0$

Here, the coefficients are $a = 1$, $b = -2$, and the constant term is $c = 0$.

Graphical Representation (Visual Verification)

The linear equation $x - 2y = 0$ can be graphed on a Cartesian coordinate system. Because the variables represent cost, the relevant domain and range are restricted to the first quadrant ($x \ge 0, y \ge 0$). The line passes through the origin $(0,0)$, confirming that if a pen costs $0$, a notebook also costs $0$.

Final Solution: The linear equation in two variables representing the given statement is $x - 2y = 0$.