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IBEW/NJATC Electrical Apprenticeship Aptitude Test (EIAT)

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2.2 Solving One-Step & Multi-Step Linear Equations

Key Takeaways

  • A linear equation represents an algebraic balance where operations performed on one side must be mirrored on the opposite side using inverse operations to isolate the variable.
  • Solving multi-step equations follows a 5-step system: expand parentheses via distribution, combine like terms, collect variable terms on one side, collect constants on the opposite side, and divide by the coefficient.
  • Equations containing fractions can be simplified instantly by multiplying every term on both sides by the Least Common Denominator (LCD) to eliminate denominators.
  • Electrical conductor voltage drop is calculated using $V_d = \frac{2KIL}{CM}$; manipulating this equation allows solving for maximum one-way distance $L$ or required wire size $CM$.
  • Always verify solved equations by substituting the calculated variable back into the original expression to confirm mathematical truth.
Last updated: August 2026

2.2 Solving One-Step & Multi-Step Linear Equations

Quick Summary: Solving linear equations is the single most tested skill on Part 1 of the EIAT. A linear equation represents a mathematical balance where both sides of an equals sign are equal. By systematically applying inverse operations, combining like terms, distributing parentheses, and clearing fractional denominators using the Least Common Denominator (LCD), candidates can isolate unknown variables ($x$) swiftly and accurately without relying on digital calculators.

Fundamental Concepts & Properties of Equality

A linear equation is an algebraic statement featuring one or more variables raised to the first power (e.g., $x^1$). Solving an equation means finding the specific value for the variable that satisfies the equation, making the left side equal to the right side.

The Balance Scale Concept

Think of an equation as a classic balance scale. Whatever operation is performed on one side of the equals sign must be performed on the opposite side to maintain balance. This balance is preserved by the fundamental Properties of Equality:

  • Addition Property of Equality: If $a = b$, then $a + c = b + c$.
  • Subtraction Property of Equality: If $a = b$, then $a - c = b - c$.
  • Multiplication Property of Equality: If $a = b$, then $a \cdot c = b \cdot c$.
  • Division Property of Equality: If $a = b$ and $c \neq 0$, then $\frac{a}{c} = \frac{b}{c}$.
                 The Balance Scale Principle
                 
                Left Side  =  Right Side
               [ 3x + 12 ] = [ 27 ]
                   |            |
             Subtract 12    Subtract 12
                   v            v
                 [ 3x ]    = [ 15 ]
                   |            |
              Divide by 3   Divide by 3
                   v            v
                  [ x ]    = [ 5 ]

Inverse Operations & Single/Two-Step Equations

To isolate a variable, apply inverse operations (operations that undo each other):

  • Addition undos Subtraction, and Subtraction undos Addition.
  • Multiplication undos Division, and Division undos Multiplication.

Single-Step Linear Equations

In single-step equations, a single inverse operation isolates the variable:

  • Example 1 (Addition/Subtraction): Solve $x - 14 = 22$. Add 14 to both sides: $x = 36$.
  • Example 2 (Multiplication/Division): Solve $6x = 42$. Divide both sides by 6: $x = 7$.

Two-Step Linear Equations

Two-step equations involve two operations. As a general rule, reverse the order of operations: undo addition and subtraction before undoing multiplication and division.

  • Example: Solve $5x + 8 = 38$.
    1. Subtract 8 from both sides: $5x = 30$.
    2. Divide both sides by 5: $x = 6$.

Multi-Step Equations: Distribution & Variables on Both Sides

Multi-step linear equations require expanding grouped expressions using the Distributive Property ($a(b + c) = ab + ac$), combining like terms on each side of the equation, and gathering all variable terms onto one side while moving all constant terms to the opposite side.

The 5-Step System for Multi-Step Equations

  1. Clear Parentheses: Apply the distributive property to eliminate all parentheses. (Be extremely careful when distributing negative signs!).
  2. Combine Like Terms on Each Side: Simplify terms on the left side, then simplify terms on the right side independently.
  3. Collect Variable Terms on One Side: Add or subtract variable terms so that all variables appear on one side of the equals sign (preferably the side that produces a positive coefficient).
  4. Collect Constant Terms on the Opposite Side: Add or subtract constant values to isolate the variable term.
  5. Isolate the Variable: Divide or multiply by the variable's coefficient to solve for $x$, then verify your answer by substituting it back into the original equation.

Clearing Fractional Denominators in Rational Equations

Equations containing fractions are common on the EIAT and frequently intimidate test takers. However, any fractional equation can be simplified instantly into an integer linear equation by multiplying every term on both sides by the Least Common Denominator (LCD).

How to Clear Fractions Step-by-Step

Consider the rational equation: x4+x−23=5\frac{x}{4} + \frac{x - 2}{3} = 5

  1. Identify the LCD: The denominators are 4 and 3. Their Least Common Denominator is $12$.
  2. Multiply Every Term by the LCD (12): 12⋅(x4)+12⋅(x−23)=12⋅(5)12 \cdot \left(\frac{x}{4}\right) + 12 \cdot \left(\frac{x - 2}{3}\right) = 12 \cdot (5)
  3. Cancel Denominators: 3x+4(x−2)=603x + 4(x - 2) = 60
  4. Distribute and Solve as an Integer Equation: 3x+4x−8=60  ⟹  7x−8=60  ⟹  7x=68  ⟹  x=6873x + 4x - 8 = 60 \implies 7x - 8 = 60 \implies 7x = 68 \implies x = \frac{68}{7}

By clearing fractions in the very first step, you eliminate complex fraction arithmetic and prevent calculation errors.

Trade Application: Voltage Drop Formula Manipulation

Electricians frequently use algebraic equation solving to calculate voltage drop over long electrical conductor runs. According to the National Electrical Code (NEC), excessive voltage drop in feeder and branch circuits causes equipment overheating, reduced motor efficiency, and inefficient power delivery.

The standard single-phase Voltage Drop Formula is:

Vd=2×K×I×LCMV_d = \frac{2 \times K \times I \times L}{CM}

Where:

  • $V_d$ = Voltage drop in Volts ($\text{V}$)
  • $K$ = Specific resistance constant of the conductor ($12.9\ \Omega\cdot\text{CM/ft}$ for copper, $21.2$ for aluminum)
  • $I$ = Circuit load current in Amperes ($\text{A}$)
  • $L$ = One-way conductor length in feet ($\text{ft}$)
  • $CM$ = Circular Mil area of the wire cross-section ($\text{CM}$)

Rearranging Formulas for Specific Variables

On the EIAT, questions often require candidates to solve the formula for an unknown variable other than $V_d$, such as maximum conductor length $L$ or required wire size $CM$:

  • Solving for Length ($L$): Multiply both sides by $CM$: $V_d \times CM = 2KIL$ Divide by $2KI$: L=Vd×CM2×K×IL = \frac{V_d \times CM}{2 \times K \times I}
  • Solving for Circular Mil Area ($CM$): CM=2×K×I×LVdCM = \frac{2 \times K \times I \times L}{V_d}

Step-by-Step Worked Examples

Worked Example 1: Multi-Step Equation with Distribution

Problem: Solve the multi-step linear equation for $x$: 4(2x−3)=3(x+6)4(2x - 3) = 3(x + 6)

Solution:

  1. Distribute Terms on Both Sides: 4⋅2x−4⋅3=3⋅x+3⋅64 \cdot 2x - 4 \cdot 3 = 3 \cdot x + 3 \cdot 6 8x−12=3x+188x - 12 = 3x + 18
  2. Subtract $3x$ from Both Sides to Collect Variable Terms: 8x−3x−12=18  ⟹  5x−12=188x - 3x - 12 = 18 \implies 5x - 12 = 18
  3. Add 12 to Both Sides to Collect Constants: 5x=18+12  ⟹  5x=305x = 18 + 12 \implies 5x = 30
  4. Divide by Coefficient 5: x=305=6x = \frac{30}{5} = 6

Verification: Substitute $x = 6$: $4(2(6) - 3) = 4(12 - 3) = 4(9) = 36$. Right side: $3(6 + 6) = 3(12) = 36$. $36 = 36$ (Correct!).

Final Answer: $x = 6$

Worked Example 2: Rational Linear Equation with LCD Clearing

Problem: Solve the fractional linear equation for $x$: x2+x3=5\frac{x}{2} + \frac{x}{3} = 5

Solution:

  1. Find LCD: Denominators are 2 and 3. The LCD is $6$.
  2. Multiply All Terms by 6: 6⋅(x2)+6⋅(x3)=6⋅(5)6 \cdot \left(\frac{x}{2}\right) + 6 \cdot \left(\frac{x}{3}\right) = 6 \cdot (5)
  3. Simplify Terms: 3x+2x=303x + 2x = 30
  4. Combine Like Terms: 5x=305x = 30
  5. Divide by 5: x=305=6x = \frac{30}{5} = 6

Final Answer: $x = 6$

Worked Example 3: Voltage Drop Conductor Length Calculation

Problem: An electrician is installing a 240V branch circuit supplying a $20\text{ A}$ load using copper conductor ($K = 12$). The allowable voltage drop $V_d$ is $6\text{ V}$. The selected wire size has a cross-sectional area of $10,400\text{ CM}$ (#10 AWG). Solve for the maximum allowable one-way circuit length $L$ in feet.

Solution:

  1. Write the Base Voltage Drop Formula: Vd=2⋅K⋅I⋅LCMV_d = \frac{2 \cdot K \cdot I \cdot L}{CM}
  2. Substitute Given Values: 6=2⋅12⋅20⋅L10,4006 = \frac{2 \cdot 12 \cdot 20 \cdot L}{10,400}
  3. Simplify Numerator Constants: 2⋅12⋅20=480  ⟹  6=480L10,4002 \cdot 12 \cdot 20 = 480 \implies 6 = \frac{480 L}{10,400}
  4. Multiply Both Sides by $10,400$: 6×10,400=480L  ⟹  62,400=480L6 \times 10,400 = 480 L \implies 62,400 = 480 L
  5. Divide by 480 to Isolate $L$: L=62,400480=6,24048=130 feetL = \frac{62,400}{480} = \frac{6,240}{48} = 130\text{ feet}

Final Answer: $130\text{ ft}$

Test Your Knowledge

Solve for $x$: $4(2x - 3) = 3(x + 6)$

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Test Your Knowledge

Clear the denominators to solve the equation: $\frac{x}{2} + \frac{x}{3} = 5$

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Test Your Knowledge

An electrician uses the voltage drop formula $V_d = \frac{2 \times K \times I \times L}{CM}$. If $V_d = 6\text{ V}$, $K = 12$, $I = 20\text{ A}$, and $CM = 10,400$, solve for maximum one-way distance $L$ in feet.

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2.3 Solving Algebraic Inequalities

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