Linear Equations in Two Variables

The One Thing

A linear equation in two variables represents a relationship between two unknown quantities, and its complete solution set forms a straight line on the Cartesian plane. Algebraic substitution, tables, graphing, intercepts, slope, and contextual modelling describe the same set of solutions in different ways.

Definitions and Results

• Linear equation in two variables: An equation that can be written as $ax+by+c=0$, where $a$, $b$, and $c$ are real numbers, and $a$ and $b$ are not both zero. Examples include $2x+3y=12$, $x-4y+5=0$, and $y=3x+1$.
• Variable: A letter or symbol representing an unknown or changeable number, such as $x$ or $y$.
• Coefficient: The numerical factor multiplying a variable, such as $a$ in $ax$ or $b$ in $by$.
• Constant: A fixed number in an equation, such as $c$ in $ax+by+c=0$.
• Ordered pair: A pair $(x,y)$, where the first number is the $x$-coordinate and the second number is the $y$-coordinate.
• Solution: An ordered pair that makes the equation true when the values of $x$ and $y$ are substituted.
• Graph: The set of all points representing the solutions of an equation on the Cartesian plane.
• Cartesian plane: A coordinate system formed by two perpendicular number lines, the coordinate axes.
• Coordinate axes: The $x$-axis is horizontal and the $y$-axis is vertical.
• Origin: The point where the coordinate axes meet, represented by $(0,0)$.
• Intercept: A point where a graph meets an axis. The $x$-intercept has $y=0$; the $y$-intercept has $x=0$.
• Slope: The rate at which $y$ changes with respect to $x$. For two points, it is the change in $y$ divided by the change in $x$.
• Graph of a linear equation: A straight line containing all ordered pairs that satisfy the equation.
• Equivalent equations: Equations with the same set of solutions, even when written in different forms.
• Contextual modelling: Using variables and equations to represent a real-life situation, such as cost, distance, age, or quantity.
• General solution property: An equation in two variables generally has infinitely many solutions because selecting one variable usually determines the other.
• Slope-intercept form: In $y=mx+d$, $m$ is the slope and $d$ is the $y$-intercept.
• Slope formula: For points $(x_1,y_1)$ and $(x_2,y_2)$, with $x_1\ne x_2$,

$$m=\frac{y_2-y_1}{x_2-x_1}.$$

• Interpretation of slope: A positive slope rises from left to right, a negative slope falls from left to right, and a zero slope produces a horizontal line.
• Vertical line: A line of the form $x=k$. Its slope is undefined, so it cannot be written as $y=mx+d$.
• One-variable line equations: $x=4$ represents a vertical line, while $y=-2$ represents a horizontal line.
• Exceptional zero-coefficient cases: $0x+0y+c=0$ is not a valid linear equation in two variables when $c\ne 0$, because it has no solution. The equation $0x+0y=0$ is true for every ordered pair, so its graph is the entire coordinate plane; this case is usually excluded from the standard definition.
• Contextual restrictions: A solution must be physically and practically meaningful. Negative, fractional, or excessively large values may need to be rejected, and units must be included. For example, $x$ may represent kilograms and $y$ may represent rupees.
• Graph interpretation: Every point on the graph is a solution, and every solution corresponds to a point on the graph.
• Practical graph restrictions: A graph may indicate non-negative quantities, maximum capacity, or a limited range of values.

Worked Methods

1. Finding solutions by assigning a value

• Choose a value for one variable.
• Substitute it into the equation.
• Calculate the corresponding value of the other variable.
• Record the result as an ordered pair.

For $x+y=5$:

• If $x=0$, then $y=5$, giving $(0,5)$.
• If $x=1$, then $y=4$, giving $(1,4)$.
• If $x=2$, then $y=3$, giving $(2,3)$.
• If $x=5$, then $y=0$, giving $(5,0)$.

An equation in two variables generally produces infinitely many such solutions.

2. Checking an ordered pair by substitution

• Substitute the $x$-coordinate and $y$-coordinate into the equation.
• Simplify both sides.
• Confirm whether the equation is true.

For $(2,3)$ and $x+y=5$:

$$2+3=5.$$

Therefore, $(2,3)$ satisfies the equation.

3. Organising solutions in a table

• Select several values of one variable.
• Calculate the corresponding values of the other variable.
• Record the ordered pairs in a table.
• Use the table to plot the points.

A table of values organises the solution set and makes graphing more systematic.

4. Graphing a linear equation

• Find at least two solution points, either by substitution or by using intercepts.
• Plot the points on the Cartesian plane.
• Join them with a straight line.
• Use a third point, if available, to verify accuracy.

For $x+y=5$, plot points such as $(0,5)$, $(1,4)$, $(2,3)$, and $(5,0)$, then join them with a straight line. The resulting line contains all ordered pairs satisfying $x+y=5$.

5. Finding intercepts

For an equation in the form $ax+by+c=0$:

• To find the $x$-intercept, put $y=0$.
• Solve for $x$.
• To find the $y$-intercept, put $x=0$.
• Solve for $y$.
• Plot the two intercepts and join them with a straight line.

The $x$-intercept lies on the $x$-axis, and the $y$-intercept lies on the $y$-axis.

6. Finding and interpreting slope

For two points $(x_1,y_1)$ and $(x_2,y_2)$, where $x_1\ne x_2$:

• Calculate the change in $y$: $y_2-y_1$.
• Calculate the change in $x$: $x_2-x_1$.
• Divide:

$$m=\frac{y_2-y_1}{x_2-x_1}.$$

• Interpret the sign:

- $m>0$: the line rises from left to right. - $m<0$: the line falls from left to right. - $m=0$: the line is horizontal.

In $y=mx+d$, identify $m$ as the slope and $d$ as the $y$-intercept.

For a vertical line $x=k$, the change in $x$ is zero, so the slope is undefined.

7. Modelling a real-life situation

• Define the variables and state their units.
• Translate the information into a linear equation.
• Solve the equation, construct a table, or graph the relationship.
• Interpret the solution in context.
• Reject values that violate practical restrictions, such as negative quantities, maximum capacity, or an invalid range.

The relationship must be linear, or reasonably approximated as linear, for a linear equation to be appropriate.

Where It Goes Wrong

• Treating any equation involving $x$ and $y$ as valid without checking the form $ax+by+c=0$, including the requirement that $a$ and $b$ are not both zero.
• Listing an ordered pair without substituting both coordinates to check that it makes the equation true.
• Plotting fewer than two solution points or failing to join the points with a straight line; a third point can be used to verify accuracy.
• Finding intercepts without setting $y=0$ for the $x$-intercept and $x=0$ for the $y$-intercept.
• Applying $y=mx+d$ to a vertical line $x=k$, whose slope is undefined.
• Accepting algebraic solutions without considering context, units, non-negative quantities, maximum capacity, or a limited range of values.

What Gets Asked

• Define a linear equation in two variables and identify its variables, coefficients, and constant.
• Determine whether a given equation, such as $2x+3y=12$, $x-4y+5=0$, or $y=3x+1$, is linear.
• Generate ordered-pair solutions by assigning values to one variable.
• Check whether a given ordered pair satisfies an equation, such as verifying that $(2,3)$ satisfies $x+y=5$.
• Complete a table of values and plot the corresponding graph.
• Explain why the graph of a linear equation is a straight line and why it contains all solutions.
• Find the $x$- and $y$-intercepts using $y=0$ and $x=0$, respectively.
• Identify slope and $y$-intercept from $y=mx+d$.
• Calculate the slope between two points using

$$m=\frac{y_2-y_1}{x_2-x_1}.$$

• Classify a line as rising, falling, horizontal, or vertical from its slope or equation.
• Distinguish between the vertical line $x=4$ and the horizontal line $y=-2$.
• Explain the exceptional cases $0x+0y+c=0$ and $0x+0y=0$.
• Form and interpret a linear equation from a real-life situation, including units and practical restrictions.
• Explain the relationship between algebraic solutions, plotted points, graphs, and equivalent equations.