Linear Equations in Two Variables

Linear Equations in Two Variables: Introduction and Formation

Extending Linear Equations to Two Variables

Previously, we studied linear equations in one variable, such as $2x + 5 = 0$. These equations involve a single unknown quantity represented by a variable and have a unique solution (provided the variable's coefficient is non-zero). Now, we expand our scope to include linear equations that involve two different variables. These equations describe a relationship between two varying quantities.

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Definition of a Linear Equation in Two Variables

A linear equation in two variables is an equation that can be written in the standard form:

In this standard form, the following conditions hold:

• $x$ and $y$: These are the two distinct variables.
• $a, b,$ and $c$: These are real coefficients (constants). They can be any real numbers.
• $a$: This is the coefficient of the variable $x$.
• $b$: This is the coefficient of the variable $y$.
• $c$: This is the constant term. It does not involve either $x$ or $y$.
• Crucially, for the equation to be considered a linear equation in two variables with a non-trivial relationship between $x$ and $y$, at least one of the coefficients of the variables, $a$ or $b$, must be non-zero. That is, $a \neq 0$ or $b \neq 0$. (This is often written concisely as $a^2 + b^2 \neq 0$). If both $a=0$ and $b=0$, the equation simplifies to $0 \cdot x + 0 \cdot y + c = 0$, or $c = 0$. If $c=0$, we get $0=0$, which is an identity (true for all $x, y$). If $c \neq 0$, we get $c=0$ (e.g., $5=0$), a contradiction (false for all $x, y$). Neither of these cases represents a relationship that defines a line in the coordinate plane.

The term "linear" indicates that the highest power of each variable is $1$, and there are no terms where the variables are multiplied together (like $xy$), or variables in denominators, or variables inside radicals.

Examples of linear equations in two variables:

• $2x + 3y - 5 = 0$: Here $a=2, b=3, c=-5$. Both $a$ and $b$ are non-zero.
• $x - 2y = 4$: This can be rearranged into the standard form by moving the constant term to the left side: $x - 2y - 4 = 0$. Here $a=1, b=-2, c=-4$. Both $a$ and $b$ are non-zero.
• $y = -3x + 1$: Rearrange to standard form: $3x + y - 1 = 0$. Here $a=3, b=1, c=-1$. Both $a$ and $b$ are non-zero.
• $5x = 7$: This can be written as $5x + 0y - 7 = 0$. Here $a=5, b=0, c=-7$. Although $b=0$, $a \neq 0$, so this is a linear equation in two variables (specifically, it represents a vertical line in the xy-plane).
• $2y = 6$: This can be written as $0x + 2y - 6 = 0$. Here $a=0, b=2, c=-6$. Although $a=0$, $b \neq 0$, so this is a linear equation in two variables (represents a horizontal line).

Examples of equations that are NOT linear equations in two variables:

• $x^2 + y = 5$: The term $x^2$ has a power of $2$.
• $xy + x = 10$: The term $xy$ involves the product of two variables.
• $\sqrt{x} + 2y = 3$: The variable $x$ is under a radical (power is $1/2$).
• $\frac{1}{x} + y = 7$: The variable $x$ is in the denominator (power is $-1$).
• $x + y + z = 1$: This involves three variables.

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Solutions of a Linear Equation in Two Variables

A solution of a linear equation in two variables $ax + by + c = 0$ is an ordered pair of real numbers $(x, y)$ that, when substituted into the equation, makes the equation a true statement (i.e., makes the LHS equal to the RHS).

Unlike linear equations in one variable which have a unique solution, a linear equation in two variables has infinitely many solutions. This is because for any value we choose for one variable (say $x$), we can find a corresponding value for the other variable (say $y$) that satisfies the equation, and vice-versa.

Example: Consider the equation $x + y = 5$. Let's find some solutions:

• If $x=0$, substitute into the equation: $0 + y = 5 \implies y=5$. The ordered pair $(0, 5)$ is a solution. Check: $0+5=5$.
• If $x=1$, substitute: $1 + y = 5 \implies y=4$. The ordered pair $(1, 4)$ is a solution. Check: $1+4=5$.
• If $x=5$, substitute: $5 + y = 5 \implies y=0$. The ordered pair $(5, 0)$ is a solution. Check: $5+0=5$.
• If $x=-1$, substitute: $-1 + y = 5 \implies y=6$. The ordered pair $(-1, 6)$ is a solution. Check: $-1+6=5$.
• If $x=2.5$, substitute: $2.5 + y = 5 \implies y=2.5$. The ordered pair $(2.5, 2.5)$ is a solution. Check: $2.5+2.5=5$.
• If $x=10$, substitute: $10 + y = 5 \implies y = 5 - 10 = -5$. The ordered pair $(10, -5)$ is a solution. Check: $10+(-5)=5$.
• If $y=0$, substitute: $x + 0 = 5 \implies x=5$. The ordered pair $(5, 0)$ is a solution. (Already found).
• If $y=3$, substitute: $x + 3 = 5 \implies x = 5 - 3 = 2$. The ordered pair $(2, 3)$ is a solution. Check: $2+3=5$.

We can continue this process indefinitely, choosing any real value for $x$ and finding a corresponding $y$, or vice-versa. Each resulting ordered pair $(x, y)$ is a solution. These infinite solutions represent all the points that lie on the straight line which is the graph of the equation (discussed in the next section).

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Forming Linear Equations in Two Variables

Linear equations in two variables are commonly used to model real-world situations where a linear relationship exists between two varying quantities. Translating a verbal description into a linear equation involves identifying the unknown quantities and expressing the relationship between them using variables.

Steps for forming a linear equation from a word problem:

1. Identify the Unknown Quantities:

   Determine the two quantities in the problem whose values are unknown or varying and which are related to each other.

2. Assign Variables:

   Choose two distinct variables (e.g., $x, y, p, q$, etc.) to represent these two unknown quantities. Clearly state what each variable represents, including units if applicable.

3. Translate the Relationship into an Equation:

   Read the problem description carefully and write an algebraic equation that expresses the relationship between the two variables and any given constants. Look for keywords that suggest equality or specific operations, as discussed in the previous section on word problems in one variable.

Answer:

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Answer:

Linear equations in two variables are fundamental tools for modelling relationships between two quantities that exhibit a constant rate of change relative to each other. Their infinite solutions are best visualized through their graphs, which are straight lines.

Graph of a Linear Equation in Two Variables

The Graphical Representation of Solutions

We've learned that a linear equation in two variables, such as $ax + by + c = 0$ (where $a$ or $b$ is non-zero), has infinitely many solutions. Each solution is a specific pair of values for $x$ and $y$ that makes the equation true, represented as an ordered pair $(x, y)$. When these ordered pairs are plotted as points on a coordinate plane (a graph with an x-axis and a y-axis), a remarkable property emerges: all these points lie on a single straight line.

Therefore, the graph of a linear equation in two variables is always a straight line.

Conversely, any straight line drawn in the coordinate plane (that is not vertical or horizontal) can be represented by a linear equation in the form $y = mx + c'$ or $ax + by + c = 0$. Vertical lines are represented by equations of the form $x = k$ (where $k$ is a constant), and horizontal lines are represented by equations of the form $y = k$ (these are special cases of $ax + by + c = 0$ where $b=0, a \neq 0$ or $a=0, b \neq 0$ respectively).

The graph provides a visual representation of the solution set of the equation. Every point $(x, y)$ that lies on the line is a solution to the equation, and every solution $(x, y)$ of the equation corresponds to a point on the line.

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Steps to Graph a Linear Equation in Two Variables

To draw the graph of a linear equation in two variables, you need to plot points that represent solutions to the equation. Since a straight line is uniquely determined by two points, finding and plotting just two solutions is theoretically enough to draw the line. However, finding a third solution is highly recommended as a check to ensure accuracy – if the three points are collinear (lie on the same straight line), you are likely correct.

Here is a systematic process for graphing a linear equation:

1. Find Solutions:

   Choose a few convenient values for one of the variables (e.g., choose a few simple integer values for $x$). Substitute each chosen value into the equation and solve for the corresponding value of the other variable (e.g., solve for $y$). This will give you a set of ordered pairs $(x, y)$ that are solutions to the equation. (Alternatively, you can choose values for $y$ and solve for $x$).

2. Organise Solutions:

   Create a table to neatly list the ordered pairs you found. Aim for at least two, and preferably three or more solutions.

3. Draw the Coordinate Plane:

   Draw the x-axis (horizontal) and the y-axis (vertical) on a piece of graph paper or a coordinate plane. Label the axes and mark a suitable scale on each axis. The origin $(0,0)$ is where the axes intersect.

4. Plot the Points:

   Plot each ordered pair $(x, y)$ from your table as a point on the coordinate plane. Remember that the first coordinate ($x$) tells you the horizontal position (right or left from the origin), and the second coordinate ($y$) tells you the vertical position (up or down from the origin).

5. Draw the Line:

   Carefully draw a straight line that passes through all the plotted points. Use a ruler to ensure it is straight. Extend the line beyond the plotted points in both directions and draw arrows on both ends to indicate that the line continues infinitely. Label the line with its equation.

A convenient way to find two points is to find the intercepts:

• To find the y-intercept, set $x=0$ in the equation and solve for $y$. The point is $(0, y_{intercept})$.
• To find the x-intercept, set $y=0$ in the equation and solve for $x$. The point is $(x_{intercept}, 0)$.

These two points are often easy to calculate and plot (unless the line passes through the origin, $(0,0)$, in which case both intercepts are at the origin and you need one more point, or if the line is horizontal/vertical, giving only one intercept).

Answer:

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Answer:

Graphing linear equations in two variables provides a visual understanding of the infinite set of solutions that satisfy the equation and their representation as a straight line in the coordinate plane.

Equations of Lines Parallel to Axes (x-axis and y-axis)

Horizontal and Vertical Lines

In the coordinate plane, lines can have various slopes and orientations. However, lines that are parallel to the coordinate axes (the x-axis and the y-axis) are special cases. They have either a slope of zero (horizontal lines, parallel to the x-axis) or an undefined slope (vertical lines, parallel to the y-axis). Their equations are simpler forms of the general linear equation in two variables because one of the variables has a constant value along the entire line.

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Equation of a Line Parallel to the x-axis (Horizontal Line)

A line parallel to the x-axis is a horizontal line. Consider any two points on such a line, say $(x_1, y_1)$ and $(x_2, y_2)$. For the line to be horizontal, its slope must be zero. The slope is calculated as $\frac{y_2 - y_1}{x_2 - x_1}$. For the slope to be zero, the numerator must be zero ($y_2 - y_1 = 0$), provided the denominator is non-zero ($x_2 \neq x_1$, which is true for any two distinct points on a non-vertical line). This implies that $y_1 = y_2$. Thus, for any two points on a horizontal line, their y-coordinates must be equal.

If a horizontal line passes through a specific point $(x_0, k)$, then every point $(x, y)$ on that line must have its y-coordinate equal to $k$. The x-coordinate can be any real number, but the y-coordinate is fixed at $k$. Therefore, the equation that describes all points on this line is simply $y = k$.

In the standard form of a linear equation $ax + by + c = 0$, if we set the coefficient of $x$ to zero ($a=0$, while $b \neq 0$), the equation becomes $0 \cdot x + by + c = 0$, which simplifies to $by + c = 0$. Solving for $y$, we get $by = -c$, or $y = -\frac{c}{b}$. Since $b$ and $c$ are constants, $-\frac{c}{b}$ is also a constant. Let $k = -\frac{c}{b}$. The equation is $y = k$, which is the equation of a horizontal line.

Example: The equation $y = 3$ represents a horizontal line located $3$ units above the x-axis. Any point on this line has a y-coordinate of $3$. Examples of points on this line are $(0, 3), (5, 3), (-10, 3)$, etc.

Example: The equation $y = -2.5$ represents a horizontal line located $2.5$ units below the x-axis.

The x-axis itself is a horizontal line that passes through the origin $(0, 0)$. For any point on the x-axis, the y-coordinate is $0$. Thus, the equation of the x-axis is $y = 0$.

The image shows a horizontal line plotted on a coordinate plane. The equation $y=k$ indicates that the y-coordinate is constant ($k$) for all points on the line, which runs parallel to the x-axis.

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Equation of a Line Parallel to the y-axis (Vertical Line)

A line parallel to the y-axis is a vertical line. Consider any two points on such a line, say $(x_1, y_1)$ and $(x_2, y_2)$. For the line to be vertical, its slope is undefined. The slope is $\frac{y_2 - y_1}{x_2 - x_1}$. For the slope to be undefined, the denominator must be zero ($x_2 - x_1 = 0$), provided the numerator is non-zero ($y_2 \neq y_1$, which is true for any two distinct points on a non-horizontal line). This implies that $x_1 = x_2$. Thus, for any two points on a vertical line, their x-coordinates must be equal.

If a vertical line passes through a specific point $(k, y_0)$, then every point $(x, y)$ on that line must have its x-coordinate equal to $k$. The y-coordinate can be any real number, but the x-coordinate is fixed at $k$. Therefore, the equation that describes all points on this line is simply $x = k$.

In the standard form of a linear equation $ax + by + c = 0$, if we set the coefficient of $y$ to zero ($b=0$, while $a \neq 0$), the equation becomes $ax + 0 \cdot y + c = 0$, which simplifies to $ax + c = 0$. Solving for $x$, we get $ax = -c$, or $x = -\frac{c}{a}$. Since $a$ and $c$ are constants, $-\frac{c}{a}$ is also a constant. Let $k = -\frac{c}{a}$. The equation is $x = k$, which is the equation of a vertical line.

Example: The equation $x = 5$ represents a vertical line located $5$ units to the right of the y-axis. Any point on this line has an x-coordinate of $5$. Examples of points on this line are $(5, 0), (5, 10), (5, -1)$, etc.

Example: The equation $x = -1.7$ represents a vertical line located $1.7$ units to the left of the y-axis.

The y-axis itself is a vertical line that passes through the origin $(0, 0)$. For any point on the y-axis, the x-coordinate is $0$. Thus, the equation of the y-axis is $x = 0$.

The image shows a vertical line plotted on a coordinate plane. The equation $x=k$ indicates that the x-coordinate is constant ($k$) for all points on the line, which runs parallel to the y-axis.

Understanding the simple forms of equations for horizontal and vertical lines ($y=k$ and $x=k$) is important for graphing and for interpreting relationships where one variable remains constant while the other varies.