Solving Linear Equations

Linear equations, where variables have a degree of 1, form straight lines when graphed. They model real-world scenarios like budgeting and motion. This MathMultiverse guide covers solving equations, systems, inequalities, and word problems with examples and visualizations.

What is a Linear Equation?

A linear equation has the form \( ax + b = c \) (single variable) or \( ax + by = c \) (two variables), where \( a \neq 0 \). Examples:

\[ 2x + 3 = 7 \] \[ 3x + 4y = 12 \]

Unlike nonlinear equations (\( x^2 + 2x + 1 = 0 \)), linear equations graph as straight lines.

Examples

Basic Equation

Solve \( 2x + 3 = 7 \):

\[ 2x = 4 \] \[ x = 2 \]

Fractions

Solve \( \frac{3x}{4} - 5 = 7 \):

\[ \frac{3x}{4} = 12 \] \[ x = 16 \]

Decimals

Solve \( 0.5x + 2.3 = 4.8 \):

\[ 0.5x = 2.5 \] \[ x = 5 \]

Variables on Both Sides

Solve \( 5x - 8 = 2x + 4 \):

\[ 3x = 12 \] \[ x = 4 \]

System of Equations

Solve:

\[ 2x + y = 5 \] \[ x - y = 1 \]

Substitution gives \( (x, y) = (2, 1) \).

Inequality

Solve \( 3x - 4 < 8 \):

\[ x < 4 \]

Word Problem

$11 for 5 items (notebooks at $3, pens at $1):

\[ n + p = 5 \] \[ 3n + p = 11 \]

Solution: 3 notebooks, 2 pens.

Visualizations

Single Equation

\( y = 2x + 3 \), intersecting \( y = 7 \):

System of Equations

\( 2x + y = 5 \), \( x - y = 1 \):

Applications

  • Budgeting: \( 2x + 3 = 7 \), buy 2 apples.
  • Physics: \( 60t + 10 = 130 \), \( t = 2 \) hours.
  • Business: \( 5x + 200 = 10x \), break-even at 40 items.
  • Geometry: Rectangle with perimeter 26, dimensions 5 and 8.
  • Mixing: 5L each of 20% and 50% solutions for 10L of 35%.
  • Investment: $4000 at 4%, $6000 at 6% yields $520.