Linear Equations, Coordinates, and Intercepts Analysis

Fundamentals of Linear Equations and Coordinates

• A linear equation in two variables represents a straight line on a two-dimensional Cartesian plane.
• The standard slope-intercept form of a linear equation is written as:   $y = mx + b$   where $m$ represents the slope of the line and $b$ represents the y-intercept.
• For the linear equation $y = 2x - 6$:
  • The slope $m$ is equal to $2$, which means that for every $1$ unit increase in $x$, the value of $y$ increases by $2$ units.
  • The y-intercept $b$ is equal to $-6$, indicating that the graph crosses the y-axis at the location where $x = 0$.

Analysis and Verification of Coordinate Points

• A coordinate pair $(x, y)$ lies on the graph of an equation if and only if substituting the values of $x$ and $y$ into the equation yields a true mathematical statement.
• Evaluating points for the equation $y = 2x - 6$:
  • When $x = 2$:     $y = 2(2) - 6 = 4 - 6 = -2$     The corresponding correct point on the graph is $(2, -2)$.
  • When $y = -4$:     $-4 = 2x - 6$$2 = 2x$$x = 1$     The corresponding correct point on the graph is $(1, -4)$.
  • When $x = 3$:     $y = 2(3) - 6 = 6 - 6 = 0$     The corresponding correct point on the graph is $(3, 0)$.
  • When $x = -2$:     $y = 2(-2) - 6 = -4 - 6 = -10$     The corresponding correct point on the graph is $(-2, -10)$.
• Algebraic verification of candidate coordinate pairs against $y = 2x - 6$:
  • Candidate $(2, -4)$: Substituting $x = 2$ gives $y = -2$. Since $-2 \neq -4$, $(2, -4)$ does not lie on the line.
  • Candidate $(-4, 1)$: Substituting $x = -4$ gives $y = 2(-4) - 6 = -14$. Since $-14 \neq 1$, $(-4, 1)$ does not lie on the line.
  • Candidate $(3, -3)$: Substituting $x = 3$ gives $y = 0$. Since $0 \neq -3$, $(3, -3)$ does not lie on the line.
  • Candidate $(-2, 6)$: Substituting $x = -2$ gives $y = -10$. Since $-10 \neq 6$, $(-2, 6)$ does not lie on the line.

Understanding and Calculating Intercepts

• An intercept is a point where a line intersects one of the coordinate axes on the Cartesian plane.
• x-intercept:
  • The x-intercept is the exact point where the line crosses the x-axis.
  • Along the x-axis, the value of the y-coordinate is identically equal to zero ($y = 0$).
  • To derive the x-intercept for $y = 2x - 6$, set $y = 0$:     $0 = 2x - 6$$2x = 6$$x = 3$
  • Expressed as a complete coordinate point, the x-intercept is $(3, 0)$.
• y-intercept:
  • The y-intercept is the exact point where the line crosses the y-axis.
  • Along the y-axis, the value of the x-coordinate is identically equal to zero ($x = 0$).
  • To derive the y-intercept for $y = 2x - 6$, set $x = 0$:     $y = 2(0) - 6$$y = -6$
  • Expressed as a complete coordinate point, the y-intercept is $(0, -6)$.