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 $y$.

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 $(-4, -1)$.

Initial Setup & Algebraic Formulation

We are tasked with finding four distinct solutions for the following linear equation in two variables:

$\pi x + y = 9$

A linear equation in two variables has infinitely many solutions. A solution is defined as an ordered pair $(x, y)$ that satisfies the equation. To systematically generate these solutions, we will express one variable in terms of the other.

Step 1: Isolating the Dependent Variable

To make the calculation of ordered pairs straightforward, we isolate the dependent variable ($y$) in terms of the independent variable ($x$). [Per the Subtraction Property of Equality, subtracting $\pi x$ from both sides of the equation yields:]

$y = 9 - \pi x$

We can now substitute arbitrary real numbers for $x$ to compute the corresponding exact values of $y$.

Step 2: Deriving the First Solution

Let us assign $x = 0$. Substituting this value into our isolated equation:

$y = 9 - \pi(0)$

$y = 9 - 0$

$y = 9$

Thus, the first solution is the ordered pair $(0, 9)$. [This point also represents the y-intercept of the linear graph].

Step 3: Deriving the Second Solution

Let us assign $x = 1$. Substituting this value into the equation:

$y = 9 - \pi(1)$

$y = 9 - \pi$

Thus, the second solution is the ordered pair $(1, 9 - \pi)$.

Step 4: Deriving the Third Solution

Let us assign $x = 2$. Substituting this value into the equation:

$y = 9 - \pi(2)$

$y = 9 - 2\pi$

Thus, the third solution is the ordered pair $(2, 9 - 2\pi)$.

Step 5: Deriving the Fourth Solution

Let us assign $x = -1$. Substituting this value into the equation:

$y = 9 - \pi(-1)$

$y = 9 + \pi$

Thus, the fourth solution is the ordered pair $(-1, 9 + \pi)$.

Step 6: Tabular Summary of Solutions

We can organize the derived solutions into a Cartesian coordinate table. This demonstrates the linear relationship and prepares the data for graphical representation.

Step 7: Graphical Representation of the Linear Equation

Below is the geometric interpretation of the equation $\pi x + y = 9$ plotted on a Cartesian plane. The line represents the locus of all infinite solutions, with our four specific calculated points highlighted. (Note: For spatial plotting purposes, $\pi$ is approximated as $3.14$).

Final Solution: Four valid solutions for the equation $\pi x + y = 9$ are $(0, 9)$, $(1, 9 - \pi)$, $(2, 9 - 2\pi)$, and $(-1, 9 + \pi)$.

Step 1: Initial Setup & Given Variables

We are given the linear equation in two variables:

$x - 2y = 4$

We need to determine whether the coordinate point $(0, 2)$ is a valid solution to this equation. In the Cartesian coordinate system, an ordered pair $(x, y)$ represents a solution to an equation if and only if substituting the $x$ and $y$ values into the equation results in a true mathematical statement [Per the Fundamental Principle of Algebraic Substitution].

• Given $x$-coordinate (abscissa): $x = 0$
• Given $y$-coordinate (ordinate): $y = 2$

Step 2: Substitution into the Left-Hand Side (LHS)

We isolate the Left-Hand Side (LHS) of the given equation and substitute the given coordinates:

$\text{LHS} = x - 2y$

Substituting $x = 0$ and $y = 2$:

$\text{LHS} = (0) - 2(2)$

$\text{LHS} = 0 - 4$

$\text{LHS} = -4$

Step 3: Comparison with the Right-Hand Side (RHS)

The Right-Hand Side (RHS) of the original equation is a constant:

$\text{RHS} = 4$

Comparing the evaluated LHS with the RHS:

$-4 \neq 4$

$\text{LHS} \neq \text{RHS}$

Because the substitution yields an inequality, the ordered pair $(0, 2)$ does not satisfy the equation.

Step 4: Geometric Interpretation & Verification

Geometrically, a linear equation in two variables represents a straight line on a Cartesian plane. Every point that lies exactly on this line is a solution to the equation. Since $(0, 2)$ does not satisfy the algebraic equation, the point $(0, 2)$ will not lie on the line $x - 2y = 4$.

To graph the line $x - 2y = 4$, we find its intercepts:

• x-intercept (set $y = 0$): $x - 2(0) = 4 \implies x = 4$. Point: $(4, 0)$
• y-intercept (set $x = 0$): $0 - 2y = 4 \implies y = -2$. Point: $(0, -2)$

The visual representation below proves that the point $(0, 2)$ is spatially disconnected from the solution set (the line).

Final Solution: Since substituting $x = 0$ and $y = 2$ results in $\text{LHS} \neq \text{RHS}$ ($-4 \neq 4$), the point $(0, 2)$ is NOT a solution to the equation $x - 2y = 4$.

Given Variables & Initial Setup

We are given a linear equation in two variables and an ordered pair representing a point on the Cartesian plane. Our objective is to determine mathematically whether the given point lies on the line defined by the equation.

• The Linear Equation: $x - 2y = 4$
• The Ordered Pair (Point): $(4, 0)$

Step 1: Coordinate Identification & Substitution

An ordered pair is written in the format $(x, y)$. [Per the fundamental principles of coordinate geometry, a point is a solution to an equation if and only if substituting its coordinates into the equation results in a true mathematical statement].

From the given point $(4, 0)$, we extract the specific coordinate values:

• Abscissa ($x$-coordinate): $x = 4$
• Ordinate ($y$-coordinate): $y = 0$

We substitute these values into the Left-Hand Side (LHS) of the given equation.

$\text{LHS} = x - 2y$

$\text{LHS} = (4) - 2(0)$

Step 2: Algebraic Evaluation

Applying the standard order of operations (PEMDAS/BODMAS), we first perform the multiplication, followed by the subtraction.

$\text{LHS} = 4 - 0$

$\text{LHS} = 4$

Step 3: Verification of Equality

We now compare the evaluated Left-Hand Side (LHS) with the Right-Hand Side (RHS) of the original equation.

• $\text{LHS} = 4$
• $\text{RHS} = 4$

Since $\text{LHS} = \text{RHS}$, the equation holds true. [By the Axiom of Equality, the substitution satisfies the linear equation, proving that the point geometrically lies exactly on the line].

Graphical Representation & Verification

To rigorously verify this result, we can map the equation $x - 2y = 4$ and the point $(4, 0)$ onto a Cartesian coordinate system. The line intersects the x-axis precisely at the point $(4, 0)$.

Final Solution: The ordered pair $(4, 0)$ satisfies the equation $x - 2y = 4$. Therefore, $(4, 0)$ is a valid solution to the given linear equation.

Given Variables & Initial Setup

We are tasked with determining whether the specific coordinate pair is a valid solution to the given linear equation in two variables.

• The Linear Equation: $x - 2y = 4$
• The Coordinate Pair to Test: $(2, 0)$

[Theoretical Foundation: A coordinate pair $(x, y)$ is considered a solution to a linear equation if and only if substituting the $x$ and $y$ values into the equation yields a true mathematical statement. This requires the Left-Hand Side (LHS) of the equation to perfectly equal the Right-Hand Side (RHS) after evaluation.]

Step 1: Isolate the Left-Hand Side (LHS) and Right-Hand Side (RHS)

From the given equation $x - 2y = 4$, we separate the expression into two distinct sides for independent evaluation:

• LHS: $x - 2y$
• RHS: $4$

Step 2: Substitution of the Coordinate Values

The given point is $(2, 0)$. In the Cartesian coordinate system, an ordered pair is formatted as $(x, y)$. Therefore, we extract the following values:

• $x = 2$
• $y = 0$

We substitute these specific values into the LHS expression:

$\text{LHS} = (2) - 2(0)$

Step 3: Algebraic Evaluation

Perform the arithmetic operations according to the standard order of operations (PEMDAS/BODMAS):

$\text{LHS} = 2 - 0$

$\text{LHS} = 2$

Step 4: Comparison and Logical Conclusion

Now, we compare the evaluated LHS with the constant RHS:

• $\text{LHS} = 2$
• $\text{RHS} = 4$

Since $2 \neq 4$, we conclude that $\text{LHS} \neq \text{RHS}$.

[Geometrical Interpretation: Because the equation is not satisfied, the point $(2, 0)$ does not lie on the straight line represented by the equation $x - 2y = 4$.]

Visual Verification: Graphical Representation

Below is the precise Cartesian plot of the line $x - 2y = 4$ alongside the tested point $(2, 0)$. Notice that the line intersects the x-axis at $(4, 0)$, leaving the point $(2, 0)$ distinctly off the line.

Final Solution: Since the substitution of $x = 2$ and $y = 0$ results in $2 \neq 4$, the coordinate pair $(2, 0)$ is NOT a solution to the equation $x - 2y = 4$.

Given Variables & Initial Setup

We are tasked with determining whether the coordinate point $(1, 1)$ is a valid solution to the given linear equation in two variables. The foundational elements of our analysis are:

• The Linear Equation: $x - 2y = 4$
• The Test Coordinate: $(1, 1)$

Step 1: Deconstructing the Coordinate Pair

In the Cartesian coordinate system, an ordered pair is universally represented in the format $(x, y)$. [By the fundamental axiom of coordinate geometry], the first value corresponds to the abscissa (the $x$-coordinate) and the second value corresponds to the ordinate (the $y$-coordinate).

Extracting the values from our test coordinate $(1, 1)$:

• $x = 1$
• $y = 1$

Step 2: Substitution into the Left-Hand Side (LHS)

To verify if the point lies on the line defined by the equation, we must substitute the extracted $x$ and $y$ values into the Left-Hand Side (LHS) of the equation. [Per the algebraic property of substitution, a point is a solution if and only if it satisfies the equality].

The LHS of the equation is given by:

$\text{LHS} = x - 2y$

Substituting $x = 1$ and $y = 1$:

$\text{LHS} = (1) - 2(1)$

$\text{LHS} = 1 - 2$

$\text{LHS} = -1$

Step 3: Comparative Analysis of LHS and RHS

We now compare our evaluated LHS with the Right-Hand Side (RHS) of the original equation.

• $\text{LHS} = -1$
• $\text{RHS} = 4$

Comparing the two values:

$-1 \neq 4$

Since the Left-Hand Side does not equal the Right-Hand Side ($\text{LHS} \neq \text{RHS}$), the coordinate pair $(1, 1)$ fails to satisfy the linear equation.

Step 4: Graphical Verification

To provide rigorous geometric proof, we can visualize the linear equation $x - 2y = 4$ and the point $(1, 1)$ on a Cartesian plane. A point is a solution to an equation if and only if it lies exactly on the line representing that equation.

As demonstrated in the high-precision graph above, the blue line represents the infinite set of solutions for $x - 2y = 4$. The red point representing the coordinate $(1, 1)$ clearly lies outside the trajectory of the line, confirming our algebraic derivation geometrically.

Final Conclusion

Because the substitution of $x = 1$ and $y = 1$ results in a mathematical contradiction ($-1 \neq 4$), the point does not satisfy the equation.

Final Solution: The coordinate $(1, 1)$ is not a solution to the equation $x - 2y = 4$.

Initial Setup & Theoretical Foundation

The given mathematical statement is the equation $y = 3x + 5$. This is a linear equation in two variables, $x$ and $y$. The standard form of a linear equation in two variables is $ax + by + c = 0$. By rearranging the given equation, we obtain $-3x + y - 5 = 0$, where the coefficients are $a = -3$, $b = 1$, and the constant term is $c = -5$.

The standard options for this classical problem are:

• (i) A unique solution
• (ii) Only two solutions
• (iii) Infinitely many solutions

Step 1: Algebraic Analysis of Solutions

A "solution" to a linear equation in two variables is defined as an ordered pair $(x, y)$ that satisfies the equation. In the equation $y = 3x + 5$, $x$ acts as the independent variable and $y$ acts as the dependent variable.

[Per the fundamental properties of real numbers], for every distinct real value assigned to $x$, there exists a corresponding unique real value for $y$. Let us systematically substitute arbitrary real numbers for $x$ to observe the resulting values of $y$:

Because the set of real numbers ($\mathbb{R}$) is infinite, we can choose infinitely many values for $x$. Consequently, we will generate infinitely many corresponding values for $y$. Therefore, the equation possesses infinitely many solutions.

Step 2: Geometric Interpretation

[By the geometric definition of linear equations], any linear equation in two variables represents a straight line on a two-dimensional Cartesian coordinate system. Every single point that lies on this line represents a valid solution $(x, y)$ to the equation.

A straight line extends indefinitely in both directions. Since a line is composed of an infinite number of points, the equation representing that line must inherently have an infinite number of solutions.

Step 3: Visual Proof via Cartesian Mapping

Below is the precise geometric representation of the equation $y = 3x + 5$. The red coordinates highlight specific solutions, demonstrating that the line passes through an infinite continuum of points.

Logical Deduction

Based on both the algebraic substitution method and the geometric mapping, it is evident that the relationship $y = 3x + 5$ is not restricted to one or two specific coordinate pairs. The equation acts as a continuous function mapping the domain of all real numbers to a range of all real numbers.

Final Solution: The true option is "Infinitely many solutions". This is because for every real value of $x$, there is a corresponding unique real value of $y$, and geometrically, the equation represents a straight line composed of an infinite number of points.

Initial Setup & Given Variables

We are tasked with verifying whether the coordinate pair $(\sqrt{2}, 4\sqrt{2})$ satisfies the given linear equation in two variables. The foundational components of our analysis are:

• The Linear Equation: $x - 2y = 4$
• The Ordered Pair: $(x, y) = (\sqrt{2}, 4\sqrt{2})$

[Per the fundamental theorem of algebra and coordinate geometry, an ordered pair $(x, y)$ is a valid solution to an equation if and only if substituting the values of $x$ and $y$ into the equation results in a true mathematical statement where the Left Hand Side (LHS) equals the Right Hand Side (RHS)].

Step 1: Substitution of Coordinates

We isolate the Left Hand Side (LHS) of the equation and substitute the specific values from the ordered pair.

Given LHS: $x - 2y$

Substitute $x = \sqrt{2}$ and $y = 4\sqrt{2}$:

$\text{LHS} = (\sqrt{2}) - 2(4\sqrt{2})$

Step 2: Algebraic Evaluation

Next, we simplify the expression using the properties of real numbers and radical arithmetic.

Multiply the constant $2$ by the coefficient of the radical term $4\sqrt{2}$ [By the associative property of multiplication]:

$\text{LHS} = \sqrt{2} - 8\sqrt{2}$

Factor out the common radical term $\sqrt{2}$ [By the distributive property of multiplication over addition/subtraction]:

$\text{LHS} = (1 - 8)\sqrt{2}$

$\text{LHS} = -7\sqrt{2}$

Step 3: Comparison with Right Hand Side (RHS)

We now compare the evaluated LHS with the constant RHS of the original equation.

Since $-7\sqrt{2}$ is an irrational number approximately equal to $-9.899$, and the RHS is the rational integer $4$, the two sides are strictly unequal.

Geometric Verification

Geometrically, the equation $x - 2y = 4$ represents a straight line on the Cartesian plane. If the point $(\sqrt{2}, 4\sqrt{2})$ were a solution, it would lie exactly on this line. The precise SVG rendering below demonstrates the spatial divergence between the line and the coordinate point.

Final Solution: Since substituting $x = \sqrt{2}$ and $y = 4\sqrt{2}$ yields $-7\sqrt{2} \neq 4$, the ordered pair $(\sqrt{2}, 4\sqrt{2})$ is NOT a solution to the equation $x - 2y = 4$.

Initial Setup & Algebraic Isolation

We are given the linear equation in two variables:

$2x + y = 7$

To systematically find solutions for this equation, it is mathematically highly efficient to express one variable in terms of the other. We will isolate the dependent variable $y$ on the left side of the equation [Per the fundamental properties of equality, by subtracting $2x$ from both sides].

$y = 7 - 2x$

A "solution" to a linear equation in two variables is an ordered pair $(x, y)$ that satisfies the equation. Since this is a linear equation defined over the set of real numbers ($\mathbb{R}$), it possesses infinitely many solutions. We will select four arbitrary integer values for the independent variable $x$ to compute the corresponding values of $y$.

Step 1: Calculating the First Solution

Let us assign the value $x = 0$. Substituting this into our isolated equation:

$y = 7 - 2(0)$

$y = 7 - 0$

$y = 7$

This yields our first ordered pair: $(0, 7)$.

Step 2: Calculating the Second Solution

Let us assign the value $x = 1$. Substituting this into the equation:

$y = 7 - 2(1)$

$y = 7 - 2$

$y = 5$

This yields our second ordered pair: $(1, 5)$.

Step 3: Calculating the Third Solution

Let us assign the value $x = 2$. Substituting this into the equation:

$y = 7 - 2(2)$

$y = 7 - 4$

$y = 3$

This yields our third ordered pair: $(2, 3)$.

Step 4: Calculating the Fourth Solution

Let us assign the value $x = 3$. Substituting this into the equation:

$y = 7 - 2(3)$

$y = 7 - 6$

$y = 1$

This yields our fourth ordered pair: $(3, 1)$.

Tabular Representation of Solutions

We can organize the derived coordinates into a data table to clearly demonstrate the linear relationship between $x$ and $y$. Notice that for every $1$ unit increase in $x$, $y$ decreases by $2$ units [This represents the slope of the line, $m = -2$].

Graphical Representation (Cartesian Plane)

Plotting these four points on a Cartesian coordinate system verifies that they are collinear, forming the straight line defined by $2x + y = 7$.

Final Solution: Four distinct solutions for the linear equation $2x + y = 7$ are $(0, 7)$, $(1, 5)$, $(2, 3)$, and $(3, 1)$.