Linear equations with one variable are equations that include just one unknown and no variables with exponents greater than 1. They're often the first type of algebraic equation students encounter. The general goal is always the same: isolate the variable to find its value. ## 3-Step Framework to Solve Any Linear Equation 1. Simplify both sides of the equation (remove brackets, combine like terms) 2. Move all variable terms to one side of the equation 3. Isolate the variable using inverse operations (addition, subtraction, multiplication, division) ## Example 1: Simple Rearrangement Solve the equation: $3x = 8 - x$ Step 1: Move all x terms to one side $3x + x = 8 - x + x$ $4x = 8$ Step 2: Divide both sides by 4 $\frac{4x}{4} = \frac{8}{4}$ $x = 2$ This means the value of x that makes both sides equal is 2. ## Example 2: Formula Rearrangement (Multiple Variables) You can apply the same solving techniques to rearrange formulas with multiple variables. > [!NOTE] > **Physics Formula:** > $m = \frac{F}{a}$ > Where $m = \text{mass}$, $F = \text{force}$, and $a = \text{acceleration}$ > > To isolate $F$, multiply both sides by $a$ > $m \cdot a = F$ Now the equation is rearranged to solve for force. ## Solving Equations with Fractions Fractional linear equations can seem harder, but multiplying by the **[[lcd|LCD]] (Least Common Denominator)** clears the fractions. ### Method 1: Eliminate Fractions Using LCD Solve: $\frac{7x}{4} + \frac{1x}{2} = \frac{63}{20}$ Step 1: Identify the LCD of 4, 2, and 20 → LCD = 20 Step 2: Multiply both sides by 20 $20\left(\frac{7x}{4} + \frac{1x}{2}\right) = 20\left(\frac{63}{20}\right)$ $35x + 10x = 63$ Step 3: Combine like terms and solve $45x = 63$ $x = \frac{63}{45} = \frac{7}{5}$ ### Method 2: Combine Like Terms with Fraction Addition $\frac{7x}{4} + \frac{1x}{2} = \frac{63}{20}$ Step 1: Convert to common denominator on the left side $\frac{1x}{2} = \frac{2x}{4}$ $\Rightarrow \frac{9x}{4} = \frac{63}{20}$ Step 2: Cross-multiply $9x \cdot 20 = 4 \cdot 63$ $180x = 252$ $x = \frac{252}{180} = \frac{7}{5}$ ## Solving Equations with Decimals When decimals are present, remove them by multiplying through with a power of 10. Solve: $0.2x + 0.4 = 1.2$ Step 1: Multiply both sides by 10 $10(0.2x + 0.4) = 10(1.2)$ $2x + 4 = 12$ Step 2: Solve the new equation $2x = 12 - 4$ $2x = 8$ $x = \frac{8}{2} = 4$ ## Evaluating and Verifying Your Solution > [!TIP] > Always check your solution by plugging it back into the original equation. Example: Original: $3x = 8 - x$ Solution: $x = 2$ Check: $3(2) = 8 - 2$ $6 = 6$ → The solution is correct. ## Practice Tests - [[Solving+Linear+Equations+Bonus+Practice+Test+1.pdf]] - [[Multi-Step+Linear+Equations+Practice+Test.pdf]]