Linear equations and graphs Knowledge-Based Study

Study notes

1. What is a Linear Equation in Two Variables?

   A linear equation in two variables is an algebraic equation that can be written in the form $Ax + By = C$, where $A$, $B$, and $C$ are real numbers, and $A$ and $B$ are not both zero.

   The graph of any linear equation in two variables is always a straight line on the coordinate plane.

   Ordered Pairs as Solutions

   A solution to a linear equation in two variables is an ordered pair $(x, y)$ that makes the equation a true statement when substituted into the variables.

   • Consider the equation $2x + y = 8$.
   • If we test the point $(3, 2)$, we substitute $x = 3$ and $y = 2$:
     $$2(3) + 2 = 6 + 2 = 8$$Since the equation holds true, the point $(3, 2)$ is a solution and lies on the graph of the line.
   • If we test $(1, 4)$:
     $$2(1) + 4 = 6 \neq 8$$The point $(1, 4)$ is not a solution and does not lie on the line.

   Every single point on the line corresponds to a solution of the equation, meaning that a line represents an infinite set of solutions.

   The Meaning of Slope

   The slope of a line measures its steepness and direction. It is defined as the ratio of the vertical change (the rise) to the horizontal change (the run) between any two distinct points on the line.

   The Slope Formula

   Given two points $(x_1, y_1)$ and $(x_2, y_2)$ on a line where $x_1 \neq x_2$, the slope $m$ is calculated as:

   $$m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}$$

   Four Types of Slopes

   • Positive Slope ($m > 0$): The line rises from left to right.
   • Negative Slope ($m < 0$): The line falls from left to right.
   • Zero Slope ($m = 0$): The line is completely horizontal (equation form: $y = c$).
   • Undefined Slope: The line is vertical because $\Delta x = 0$, causing division by zero (equation form: $x = k$).

   Common Algebraic Forms

   Linear equations can be written in several equivalent forms depending on the information given or needed.

   1. Slope-Intercept Form

   The slope-intercept form is expressed as:

   $$y = mx + b$$

   In this equation:

   • $m$ is the slope of the line.
   • $b$ is the y-intercept, which is the point $(0, b)$ where the line crosses the y-axis.

   2. Point-Slope Form

   When a specific point $(x_1, y_1)$ and the slope $m$ are known, we use:

   $$y - y_1 = m(x - x_1)$$

   3. Standard Form

   The standard form is written as:

   $$Ax + By = C$$

   where $A$, $B$, and $C$ are typically integers, and $A \geq 0$. This form is especially useful for finding x- and y-intercepts quickly.

   Techniques for Graphing

   To draw the graph of a line, we need at least two points. Two common methods make this process straightforward.

   Method 1: Using Slope and Y-Intercept

   1. Identify the y-intercept $b$ from $y = mx + b$ and plot the point $(0, b)$.
   2. Use the slope $m = \frac{\text{rise}}{\text{run}}$ from $(0, b)$ to find a second point. Move vertically by the rise and horizontally by the run.
   3. Draw a straight line passing through both points and extend it with arrows in both directions.

   Method 2: Using the Intercepts

   This method is ideal for equations in standard form $Ax + By = C$:

   • Find the x-intercept: Set $y = 0$ and solve for $x$. Plot $(x, 0)$.
   • Find the y-intercept: Set $x = 0$ and solve for $y$. Plot $(0, y)$.
   • Connect the two intercepts with a straight line.

   Complete Worked Example

   Problem: Graph the linear equation $3x - 2y = 6$ and identify its slope and intercepts.

   Step 1: Find the Intercepts

   • To find the x-intercept, let $y = 0$:
     $$3x - 2(0) = 6 \implies 3x = 6 \implies x = 2$$The x-intercept is $(2, 0)$.
   • To find the y-intercept, let $x = 0$:
     $$3(0) - 2y = 6 \implies -2y = 6 \implies y = -3$$The y-intercept is $(0, -3)$.

   Step 2: Convert to Slope-Intercept Form

   Isolate $y$ to find the slope:

   $$3x - 2y = 6$$$$-2y = -3x + 6$$$$y = \frac{3}{2}x - 3$$

   Here, the slope is $m = \frac{3}{2}$ and the y-intercept is $(0, -3)$, confirming our previous calculation.

   Step 3: Sketch the Graph

   Plot $(0, -3)$, move up 3 units and right 2 units to arrive at $(2, 0)$, and draw the continuous straight line passing through them.