Linear Equations, Slope-Intercept Form, and Intercepts

Point and Slope Identification

• A given point on a line is represented as $(-2, -6)$.

• The given slope of the line is $m = -4$

Converting Linear Equations to Slope-Intercept Form

Slope-intercept form of a linear equation is written as $y = mx + b$, where $m$ represents the slope and $b$ represents the $y$-intercept.

• Conversion of Equation 1: $9x - 6y = -90$

  • Step 1: Subtract $9x$ from both sides of the equation to isolate the $y$-term:     $9x - 6y - 9x = -90 - 9x$     $-6y = -9x - 90$

  • Step 2: Divide both sides by $-6$ to solve for $y$:     $\frac{-6y}{-6} = \frac{-9x - 90}{-6}$

  • Step 3: Simplify fractions:     $y = \frac{9}{6}x + \frac{90}{6}$     $y = \frac{3}{2}x + 15$

  • The slope ($m$) derived from this line is $\frac{3}{2}$.

• Conversion of Equation 2: $x + 6y = -90$

  • Step 1: Isolate the term containing $y$:     $6y = x - 90$

  • Step 2: Divide both sides by $6$:     $\frac{6y}{6} = \frac{x - 90}{6}$

  • Step 3: Simplify into slope-intercept form:     $y = \frac{1}{6}x - 15$

Perpendicular Slopes

• Two non-vertical lines are perpendicular if their slopes are negative reciprocals of each other.

• Formula for perpendicular slope ($m_{\perp}$):   $m_{\perp} = -\frac{1}{m}$

• For the line $y = \frac{3}{2}x + 15$ with slope $m = \frac{3}{2}$:

  • Taking the reciprocal yields $\frac{2}{3}$.

  • Negating the reciprocal yields the perpendicular slope:     $m_{\perp} = -\frac{2}{3}$

Finding Intercepts of Linear Equations

• General Rules for Intercepts:

  • To find the $x$-intercept ($x\text{-int}$), set $y = 0$ and solve for $x$.

  • To find the $y$-intercept ($y\text{-int}$), set $x = 0$ and solve for $y$.

• Example 1: $3x - 5y = 30$

  • Finding the $x$-intercept:

  • Set $y = 0$:       $3x - 5(0) = 30$       $3x = 30$       $\frac{3x}{3} = \frac{30}{3}$       $x = 10$

  • $x$-intercept coordinate: $(10, 0)$

  • Finding the $y$-intercept:

  • Set $x = 0$:       $3(0) - 5y = 30$       $-5y = 30$       $\frac{-5y}{-5} = \frac{30}{-5}$       $y = -6$

  • $y$-intercept coordinate: $(0, -6)$

• Example 2: $-3x + y = 26$

  • Finding the $x$-intercept:

  • Set $y = 0$:       $-3x + (0) = 26$       $-3x = 26$       $x = -\frac{26}{3}$

  • $x$-intercept coordinate: $\left(-\frac{26}{3}, 0\right)$

  • Finding the $y$-intercept:

  • Set $x = 0$:       $-3(0) + y = 26$       $y = 26$

  • $y$-intercept coordinate: $(0, 26)$

• Example 3: $-x - 2y = 3$

  • Finding the $x$-intercept:

  • Set $y = 0$:       $-x - 2(0) = 3$       $-x = 3$       $x = -3$

  • $x$-intercept coordinate: $(-3, 0)$

  • Finding the $y$-intercept:

  • Set $x = 0$:       $-0 - 2y = 3$       $-2y = 3$       $y = -\frac{3}{2}$

  • $y$-intercept coordinate: $\left(0, -\frac{3}{2}\right)$