Solving Linear Equations

Linear equations are fundamental in algebra, representing relationships where variables have a degree of 1, resulting in straight-line graphs. They are essential for modeling real-world scenarios such as budgeting, motion, and business calculations. This guide will explore solving linear equations through various methods, including single equations, systems of equations, inequalities, and word problems, with step-by-step examples, graphical insights, and practical applications to demonstrate their utility.

What is a Linear Equation?

A linear equation is an equation where the highest power of the variable is 1. It can be written in the form \( ax + b = c \) for a single variable, or \( ax + by = c \) for two variables, where \( a \), \( b \), and \( c \) are constants, and \( a \neq 0 \). When graphed, it forms a straight line, which is why it’s called "linear." For example:

\[ 2x + 3 = 7 \]

Here, \( x \) is the variable, and solving for \( x \) means finding the value that makes the equation true. Linear equations can also involve multiple variables, such as:

\[ 3x + 4y = 12 \]

In this case, the equation represents a line in a 2D plane, and solving it often involves finding pairs \( (x, y) \) that satisfy the equation, typically as part of a system of equations. Linear equations are distinct from nonlinear equations (e.g., \( x^2 + 2x + 1 = 0 \)), which involve higher powers and graph as curves.

Examples: Solving Equations, Systems, Inequalities, Word Problems

Let’s dive into solving linear equations with various scenarios, breaking down each step for clarity.

Example 1: Basic Linear Equation

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

  1. Isolate the variable term: Subtract 3 from both sides.
    2. \[ 2x + 3 - 3 = 7 - 3 \] \[ 2x = 4 \]
  2. Solve for \( x \): Divide both sides by 2.
    3. \[ \frac{2x}{2} = \frac{4}{2} \] \[ x = 2 \]
  3. Verify: Substitute \( x = 2 \).
    4. \[ 2(2) + 3 = 4 + 3 \] \[ = 7 \]

The solution \( x = 2 \) is correct.

Example 2: Equation with Fractions

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

  1. Isolate the variable term: Add 5 to both sides.
    2. \[ \frac{3x}{4} - 5 + 5 = 7 + 5 \] \[ \frac{3x}{4} = 12 \]
  2. Eliminate the fraction: Multiply both sides by 4.
    3. \[ 4 \times \frac{3x}{4} = 4 \times 12 \] \[ 3x = 48 \]
  3. Solve for \( x \): Divide both sides by 3.
    4. \[ \frac{3x}{3} = \frac{48}{3} \] \[ x = 16 \]
  4. Verify: Substitute \( x = 16 \).
    5. \[ \frac{3(16)}{4} - 5 = \frac{48}{4} - 5 \] \[ = 12 - 5 \] \[ = 7 \]

The solution is \( x = 16 \).

Example 3: Equation with Decimals

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

  1. Isolate the variable term: Subtract 2.3 from both sides.
    2. \[ 0.5x + 2.3 - 2.3 = 4.8 - 2.3 \] \[ 0.5x = 2.5 \]
  2. Solve for \( x \): Divide both sides by 0.5 (or multiply by 2).
    3. \[ \frac{0.5x}{0.5} = \frac{2.5}{0.5} \] \[ x = 5 \]
  3. Verify: Substitute \( x = 5 \).
    4. \[ 0.5(5) + 2.3 = 2.5 + 2.3 \] \[ = 4.8 \]

The solution is \( x = 5 \).

Example 4: Variables on Both Sides

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

  1. Move variable terms to one side: Subtract \( 2x \) from both sides.
    2. \[ 5x - 2x - 8 = 2x - 2x + 4 \] \[ 3x - 8 = 4 \]
  2. Isolate the variable term: Add 8 to both sides.
    3. \[ 3x - 8 + 8 = 4 + 8 \] \[ 3x = 12 \]
  3. Solve for \( x \): Divide both sides by 3.
    4. \[ \frac{3x}{3} = \frac{12}{3} \] \[ x = 4 \]
  4. Verify: Substitute \( x = 4 \).
    5. \[ 5(4) - 8 = 20 - 8 = 12 \] \[ 2(4) + 4 = 8 + 4 = 12 \]

The solution is \( x = 4 \).

Example 5: System of Linear Equations

Solve the system using substitution:

\[ 2x + y = 5 \] \[ x - y = 1 \]
  1. Solve the second equation for \( x \):
    2. \[ x - y = 1 \] \[ x = y + 1 \]
  2. Substitute into the first equation:
    3. \[ 2(y + 1) + y = 5 \] \[ 2y + 2 + y = 5 \] \[ 3y + 2 = 5 \]
  3. Solve for \( y \): Subtract 2, then divide by 3.
    4. \[ 3y + 2 - 2 = 5 - 2 \] \[ 3y = 3 \] \[ y = 1 \]
  4. Find \( x \): Use \( x = y + 1 \).
    5. \[ x = 1 + 1 \] \[ x = 2 \]
  5. Verify: Check both equations.
    6. \[ 2(2) + 1 = 4 + 1 = 5 \] \[ 2 - 1 = 1 \]

Solution: \( (x, y) = (2, 1) \).

Example 6: Linear Inequality

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

  1. Isolate the variable term: Add 4 to both sides.
    2. \[ 3x - 4 + 4 < 8 + 4 \] \[ 3x < 12 \]
  2. Solve for \( x \): Divide both sides by 3.
    3. \[ \frac{3x}{3} < \frac{12}{3} \] \[ x < 4 \]
  3. Verify: Test \( x = 3 \).
    4. \[ 3(3) - 4 = 9 - 4 = 5 \] \[ 5 < 8 \text{ (true)} \]

Solution: \( x < 4 \).

Example 7: Word Problem

A store sells notebooks for $3 each and pens for $1 each. You spend $11 on 5 items. How many notebooks and pens did you buy?

  1. Define variables: Let \( n \) be notebooks, \( p \) be pens.
  2. Set up equations:
    3. \[ n + p = 5 \] \[ 3n + p = 11 \]
  3. Solve using substitution: From the first, \( p = 5 - n \).
    4. \[ 3n + (5 - n) = 11 \] \[ 3n + 5 - n = 11 \] \[ 2n + 5 = 11 \] \[ 2n = 6 \] \[ n = 3 \]
  4. Find \( p \):
    5. \[ p = 5 - 3 \] \[ p = 2 \]
  5. Verify:
    6. \[ 3 + 2 = 5 \] \[ 3(3) + 2 = 9 + 2 = 11 \]

You bought 3 notebooks and 2 pens.

Graphical Insight

Linear equations graph as straight lines. Let’s visualize some examples.

Graph 1: Single Equation

From Example 1, \( y = 2x + 3 \), solving \( 2x + 3 = 7 \) finds the intersection with \( y = 7 \):

The intersection at \( x = 2 \) confirms our solution.

Graph 2: System of Equations

From Example 5, graph \( 2x + y = 5 \) and \( x - y = 1 \):

Intersection at \( (2, 1) \) matches our solution.

Real-World Application

Linear equations model various scenarios. Here are detailed examples:

  • Budgeting: From Example 1, with $7 for apples ($2 each) and a $3 fee:
    \[ 2x + 3 = 7 \] \[ x = 2 \]
    You can buy 2 apples.
  • Physics - Motion: A car travels at 60 km/h with a 10 km head start. Distance after \( t \) hours: \( d = 60t + 10 \). When is \( d = 130 \)?
    \[ 60t + 10 = 130 \] \[ 60t = 120 \] \[ t = 2 \]
    After 2 hours.
  • Business - Break-Even: Cost to produce \( x \) items is \( 5x + 200 \), revenue is \( 10x \). Break-even point:
    \[ 5x + 200 = 10x \] \[ 200 = 10x - 5x \] \[ 200 = 5x \] \[ x = 40 \]
    Break-even at 40 items.
  • Geometry - Perimeter: A rectangle’s length is 3 more than its width. Perimeter is 26. Find dimensions:
    \[ l = w + 3 \] \[ 2l + 2w = 26 \] \[ 2(w + 3) + 2w = 26 \] \[ 2w + 6 + 2w = 26 \] \[ 4w + 6 = 26 \] \[ 4w = 20 \] \[ w = 5 \] \[ l = 5 + 3 = 8 \]
    Width is 5, length is 8.
  • Mixing Solutions: Mix a 20% solution with a 50% solution to get 10 liters of 35% solution:
    \[ x + y = 10 \] \[ 0.2x + 0.5y = 0.35 \times 10 \] \[ 0.2x + 0.5y = 3.5 \]
    Solve:
    \[ y = 10 - x \] \[ 0.2x + 0.5(10 - x) = 3.5 \] \[ 0.2x + 5 - 0.5x = 3.5 \] \[ -0.3x + 5 = 3.5 \] \[ -0.3x = -1.5 \] \[ x = 5 \] \[ y = 10 - 5 = 5 \]
    Use 5 liters of each.
  • Investment: $10,000 invested at 4% and 6% yields $520 annually. How much at each rate?
    \[ x + y = 10000 \] \[ 0.04x + 0.06y = 520 \]
    Solve:
    \[ y = 10000 - x \] \[ 0.04x + 0.06(10000 - x) = 520 \] \[ 0.04x + 600 - 0.06x = 520 \] \[ -0.02x + 600 = 520 \] \[ -0.02x = -80 \] \[ x = 4000 \] \[ y = 6000 \]
    $4000 at 4%, $6000 at 6%.