1. **Aljafna (Equations)** - **Title:** Algebraic Equations - **Formula / Solution:** To solve $ax + b = 0$, isolate $x$: $$x = \frac{-b}{a}$$ - **Example:** Solve $3x + 6 = 0$ \nSteps: 1. Subtract 6 from both sides: $3x = -6$ 2. Divide both sides by 3: $x = \frac{-6}{3} = -2$ \n2. **Einangra breytu úr formúlu (Isolate variable from a formula)** - **Title:** Isolating a Variable - **Formula / Solution:** Given $A = 2bx + 3y$, isolate $x$: $$x = \frac{A - 3y}{2b}$$ - **Example:** Isolate $x$ from $A = 2bx + 3y$ \nSteps: 1. Subtract $3y$: $A - 3y = 2bx$ 2. Divide by $2b$: $x = \frac{A - 3y}{2b}$ \n3. **Jöfnuhneppi með samlagningaraðferð (System of equations by addition method)** - **Title:** System of Equations - Addition - **Formulas:** For $$\begin{cases} 2x + 3y = 7 \\ x - 3y = 1 \end{cases}$$ Adding eliminates $y$. - **Example:** Solve the above system \nSteps: 1. Add: $(2x + 3y) + (x - 3y) = 7 + 1$ 2. Simplify: $3x = 8$ 3. Solve for $x$: $x = \frac{8}{3}$ 4. Substitute in second eq: $\frac{8}{3} - 3y = 1$ 5. Solve for $y$: $-3y = 1 - \frac{8}{3} = \frac{3}{3} - \frac{8}{3} = -\frac{5}{3}$ 6. $y = \frac{5}{9}$ \n4. **Jöfnuhneppi með innsetningaraðferð (System of equations by substitution)** - **Title:** System of Equations - Substitution - **Formulas:** Using the same system $$\begin{cases} 2x + 3y = 7 \\ x - 3y = 1 \end{cases}$$ Solve second for $x$ and substitute. - **Example:** Use substitution \nSteps: 1. From $x - 3y = 1$, get $x = 1 + 3y$ 2. Substitute in first equation: $$2(1 + 3y) + 3y = 7$$ 3. Expand: $2 + 6y + 3y = 7$ 4. Combine: $2 + 9y = 7$ 5. Subtract 2: $9y = 5$ 6. Solve: $y = \frac{5}{9}$ 7. Substitute $y$ back to get $x$: $x = 1 + 3 \times \frac{5}{9} = 1 + \frac{15}{9} = \frac{24}{9} = \frac{8}{3}$ \n5. **Finna skurðpunkt við y- og x-ás (Find intercepts with y- and x-axis)** - **Title:** Intercepts - **Formula / Solution:** - $x$-intercept: set $y=0$, solve for $x$ - $y$-intercept: set $x=0$, solve for $y$ - **Example:** For $y = 2x + 3$ \nSteps: 1. $x$-intercept ($y=0$): $0 = 2x + 3$, $2x = -3$, $x = -\frac{3}{2}$ 2. $y$-intercept ($x=0$): $y = 2(0) + 3 = 3$ \n6. **Domain** - **Title:** Domain of a function - **Explanation:** Set of all $x$ values for which the function is defined. - **Example:** For $f(x) = \frac{1}{x-2}$, domain excludes $x=2$ \nSteps: 1. Identify values that make denominator zero: $x=2$ 2. Domain: all real numbers except $2$ \n7. **Range** - **Title:** Range of a function - **Explanation:** Set of all $y$ values the function can take. - **Example:** For $y = x^2$, range is $y \geq 0$ \nSteps: 1. Since $x^2$ is always non-negative, minimum value is 0 2. Range: all real $y \geq 0$