Notes on Graphing Linear Functions and Systems of Linear Equations

Graphing linear functions and the rectangular coordinate system

• Basic components of the rectangular coordinate system:
  • Axes: x-axis (horizontal) and y-axis (vertical)
  • Origin at \((0,0)\)
  • Points denoted as \((x,y)\)
• Linear relationships yield straight lines when graphed on the coordinate plane.
• Forms of linear equations:
  • Slope-intercept form: \(y = mx + b\) where \(m\) is the slope (rate of change) and \(b\) is the y-intercept.
  • Standard form: \(Ax + By = C\) where \(A,B\) are not both zero.
  • Vertical lines occur when the equation reduces to \(x = c\).
  • Horizontal lines occur when the equation reduces to \(y = d\).
• Intercepts:
  • x-intercept: set \(y = 0\) and solve for x; in standard form this is \(x = C/A\) (if \(A\neq 0\)).
  • y-intercept: set \(x = 0\) and solve for y; in standard form this is \(y = C/B\) (if \(B\neq 0\)).
• Key concept: a linear equation in two variables represents a line; the line is the set of all points satisfying the equation.
• Example problems from the transcript:
  • Example 1:
  • Equation: \-2x + 3 = 4\
  • Solve for x: \-2x = 1 \Rightarrow x = -\frac{1}{2}\
  • Graph interpretation: vertical line at \(x = -\frac{1}{2}\) (every y is allowed since y is not present).
  • Example 2:
  • Equation: \-2x + 3 = 0\
  • Solve for x: \-2x = -3 \Rightarrow x = \frac{3}{2}\
  • Graph interpretation: vertical line at \(x = \frac{3}{2}\).
  • Example 3:
  • Equation: \-2x + 3 = 1\
  • Solve for x: \-2x = -2 \Rightarrow x = 1\
  • Graph interpretation: vertical line at \(x = 1\).
  • Example 4 (two-variable linear equation):
  • Equation: \(2x + 5y = -4\
  • Solve for y (slope-intercept form): \y = -\frac{2}{5}x - \frac{4}{5}\
  • Slope: \(m = -\frac{2}{5}\)
  • y-intercept: \(b = -\frac{4}{5}\)
  • Intercepts:
    • x-intercept (set y=0): \(2x = -4 \Rightarrow x = -2\)
    • y-intercept (set x=0): \(y = -\frac{4}{5}\)
  • Check a solution: \( (3,-2) \
    \) satisfies the equation since \(2(3) + 5(-2) = 6 - 10 = -4\).
  • Graph description: line passing through \((-2,0)\) and \(0,-\frac{4}{5}\)
• Worked set notation example from the transcript:
  • The set of points solving \-2x + 3 = 1\ is a vertical line consisting of all points with \(x = 1\), i.e., \{(x,y) | x = 1\}.
• Graphing approach best practices:
  • When the equation has no y-term (only x), it represents a vertical line \(x = c\).
  • When the equation has both x and y, convert to slope-intercept form to read off the slope and intercepts, or plot two points and draw the line.
  • Use intercepts to quickly sketch the graph: plot \(x-intercept\) and \(y-intercept\), then draw the line through them.
• Substitution and verification:
  • You can verify a solution by substitution into the original equation(s).
  • Example verification: for \(2x + 5y = -4\), the point \( (3,-2) \
    \) yields \(2(3) + 5(-2) = -4\).
• Connections to broader concepts:
  • Linear equations model constant-rate relationships: distance vs. time, cost vs. quantity, etc.
  • The slope represents the rate of change, and the intercept represents the starting value when the other variable is zero.
• Practical implications:
  • Misidentifying the form (e.g., treating a vertical line as a function y = f(x)) leads to incorrect conclusions; vertical lines are not functions in the traditional sense.
  • When graphing real-world data, ensure units and scales are consistent to avoid misinterpretation.
• Quick reference formulas:
  • From Ax + By = C: \text{Slope } m = -\frac{A}{B}, \text{x-intercept } x = \frac{C}{A} \ (if A \neq 0), \text{y-intercept } y = \frac{C}{B} \ (if B \neq 0).
  • Slope-intercept form: \(y = mx + b\).
  • For two-variable linear equation: \y = -\frac{A}{B}x + \frac{C}{B}\ when B \neq 0.
• Summary takeaway:
  • Graphs of linear equations in two variables are straight lines.
  • Vertical lines occur when the equation lacks a y-term (x is constant).
  • Two-point plots and intercepts are practical methods for sketching lines quickly.