Comprehensive Algebra Review: Linear Equations, Inequalities, Graphing, and Systems

Linear Equations and Inequalities

1. Solving Single-Variable Equations

   • Isolate the variable $x$ using inverse operations.

   • Example: Solve $-6x - 5 = -13$

     • Add $5$ to both sides: $-6x = -8$

     • Divide by $-6$: $x = \frac{4}{3}$

2. Solving Literal Equations

   • Rearrange algebraic formulas for a specified target variable.

   • Example 1: Solve for $m$ in $y = mx + b$

     • Subtract $b$ from both sides: $y - b = mx$

     • Divide by $x$: $m = \frac{y - b}{x}$

   • Example 2: Solve for $y$ in $9x - 3y = 15$

     • Subtract $9x$: $-3y = -9x + 15$

     • Divide by $-3$: $y = 3x - 5$

3. Solving and Graphing Linear Inequalities

   • Remember to flip the inequality sign when multiplying or dividing both sides by a negative number.

   • Example 1: Solve $5x - 4 \le 16$

     • Add $4$: $5x \le 20$

     • Divide by $5$: $x \le 4$

   • Example 2: Solve $-3x + 1 < 2x + 11$

     • Subtract $2x$: $-5x + 1 < 11$

     • Subtract $1$: $-5x < 10$

     • Divide by $-5$ and reverse inequality symbol: $x > -2$

Graphing Linear Equations

1. Graphing Using a Table of Values

   • Select sample inputs for $x$ and compute outputs for $y = 3x - 1$.

2. Graphing Using Intercepts

   • Equation: $-4x + 3y = 24$

     • Calculate $x$-intercept (set $y = 0$): $-4x = 24 \Rightarrow x = -6$, giving point $(-6, 0)$

     • Calculate $y$-intercept (set $x = 0$): $3y = 24 \Rightarrow y = 8$, giving point $(0, 8)$

3. Graphing Using Slope and y-Intercept

   • Equation: $y = -3x + 1$

     • Identify slope $m = -3$

     • Identify $y$-intercept $(0, 1)$

Writing Linear Equations

1. Given Slope and a Point

   • Find equation with slope $m = -2$ containing point $(-1, 1)$

   • Use point-slope form $y - y<em>1 = m(x - x</em>1)$:

     • $y - 1 = -2(x - (-1))$

     • $y - 1 = -2x - 2$

     • $y = -2x - 1$

2. Given Two Points

   • Find equation containing $(3, 2)$ and $(4, 7)$

   • Calculate slope: $m = \frac{7 - 2}{4 - 3} = 5$

   • Use point-slope form:

     • $y - 2 = 5(x - 3)$

     • $y = 5x - 13$

3. Perpendicular Lines

   • Find equation of line perpendicular to $y = -2x$ passing through $(2, 3)$

   • The negative reciprocal slope is $m = \frac{1}{2}$

   • Use point-slope form:

     • $y - 3 = \frac{1}{2}(x - 2)$

     • $y = \frac{1}{2}x + 2$

Systems of Equations

1. Graphing Linear Inequalities in Two Variables

   • Boundary lines:

     • $3x + 4y = 12$

     • $-3x + 2y = 6$

   • Graph boundary lines and shade appropriate regions based on inequalities $3x + 4y \le 12$ and $-3x + 2y > 6$.

2. Solving Systems by Substitution

   • System: $3x + 4y = -4$ and $y = 4x + 17$

   • Substitute $y$ into first equation:

     • $3x + 4(4x + 17) = -4$

     • $19x + 68 = -4$

     • $19x = -72 \Rightarrow x = -\frac{72}{19}$

   • Substitute $x$ back to determine $y$.

3. Solving Systems by Elimination

   • System: $-3x + 2y = -5$ and $2x - y = 10$

   • Multiply second equation by $2$: $4x - 2y = 20$

   • Add equations together:

     • $(-3x + 4x) + (2y - 2y) = -5 + 20$

     • $x = 15$

   • Substitute $x = 15$ to solve for $y$:

     • $2(15) - y = 10 \Rightarrow y = 20$

4. Three-Variable Systemsq

   • System of equations:

     • $x - 3y + 2z = 13$

     • $3x + y - 4z = 13$

     • $-4x - 4y + 2z = 0$

   • Use elimination or substitution systematically to isolate and solve for $x$, $y$, and $z$.