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Solve linear equations in one or more variables using various algebraic techniques. Understand their properties.

Straight-Line Solutions: Linear Equations

Linear equations are equations where the highest power of the variable is 1, producing a straight line when graphed. Solving them means finding the value of the variable that makes both sides equal, using inverse operations to isolate the variable step by step. Linear equations model countless real-world relationships β€” distance over time, cost versus quantity, and temperature conversions.

Components of Linear Equations

This section covers the key techniques for solving linear equations:

  • Multi-Step Equations: Equations requiring multiple inverse operations, working in reverse order of operations to isolate the variable.
  • Variables on Both Sides: Moving variable terms to one side and constants to the other by adding or subtracting from both sides.
  • Equations with Fractions & Decimals: Clearing fractions by multiplying both sides by the LCD, or clearing decimals by multiplying by a power of 10.
  • Slope-Intercept Form: Writing linear equations as y = mx + b, where m is the slope and b is the y-intercept.

Examples of Linear Equations

Multi-Step Equation Examples

  • Solve 4x - 7 = 13: Add 7 to both sides to get 4x = 20, then divide by 4 to get x = 5.
  • Solve 3(x + 2) = 21: Distribute to get 3x + 6 = 21, subtract 6 to get 3x = 15, divide by 3 to get x = 5.
  • Solve -2x + 9 = 1: Subtract 9 to get -2x = -8, divide by -2 to get x = 4.

Variables on Both Sides Examples

  • Solve 5x + 3 = 2x + 18: Subtract 2x from both sides to get 3x + 3 = 18, subtract 3 to get 3x = 15, divide by 3 to get x = 5.
  • Solve 7n - 4 = 3n + 12: Subtract 3n to get 4n - 4 = 12, add 4 to get 4n = 16, divide by 4 to get n = 4.
  • Solve 2(y + 5) = y + 16: Distribute to get 2y + 10 = y + 16, subtract y to get y + 10 = 16, subtract 10 to get y = 6.

Equations with Fractions Examples

  • Solve x/3 + 2 = 5: Subtract 2 to get x/3 = 3, multiply by 3 to get x = 9.
  • Solve 2x/5 - 1 = 3: Add 1 to get 2x/5 = 4, multiply by 5 to get 2x = 20, divide by 2 to get x = 10.
  • Solve 0.3x + 1.5 = 4.5: Multiply everything by 10 to clear decimals: 3x + 15 = 45, subtract 15 to get 3x = 30, so x = 10.

Slope-Intercept Form Examples

  • A line with slope 2 and y-intercept -3 has equation y = 2x - 3. When x = 4, y = 2(4) - 3 = 5.
  • Find the equation of a line passing through (0, 5) with slope -1: Since b = 5 and m = -1, the equation is y = -x + 5.
  • A phone plan costs $25 per month plus $0.10 per text. The equation is y = 0.10x + 25, where x is texts and y is total cost.