3.2 Solving Linear Equations in One Variable

Core Fundamentals of Linear Equations

A linear equation in one variable is an algebraic statement asserting that two expressions containing a single variable raised to the first power ($x^1$) are equal. The standard algebraic form of a linear equation in one variable is:

$$ax + b = c$$

where $a$, $b$, and $c$ are real numbers and $a \neq 0$. To solve a linear equation means to determine the exact numerical value of the variable (called the solution or root) that makes the equation a true mathematical statement.

In healthcare environments, nurses continuously solve single-variable linear equations—often mental or written—to determine exact drug infusion volumes, adjust IV pump rates, and calculate pediatric weight-based medication dosages. A thorough mastery of linear equation isolation procedures ensures speed, confidence, and absolute mathematical precision on the TEAS 7 exam.

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The Addition and Multiplication Properties of Equality

Solving a linear equation relies on performing inverse operations to isolate the variable on one side of the equals sign. The two fundamental properties governing equation manipulation are:

1. Addition Property of Equality: If $a = b$, then $a + c = b + c$ and $a - c = b - c$. You can add or subtract any number from both sides without changing the equality.
2. Multiplication Property of Equality: If $a = b$, then $a \cdot c = b \cdot c$ and $\frac{a}{c} = \frac{b}{c}$ (provided $c \neq 0$). You can multiply or divide both sides by any non-zero number without changing the equality.

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Master 5-Step Algorithm for Solving Multi-Step Equations

When encountering complex linear equations on the TEAS 7 featuring parentheses, fractions, and variables on both sides, follow this universal 5-step algorithm:

[Step 1: Clear Fractions/Decimals] --> Multiply all terms by the LCD
[Step 2: Distribute]                --> Remove all parentheses
[Step 3: Combine Like Terms]        --> Simplify left and right sides independently
[Step 4: Collect Variable Terms]   --> Move all variable terms to one side
[Step 5: Isolate Variable]          --> Undo remaining addition/subtraction, then division

Step-by-Step Worked Example 1: Variables on Both Sides with Parentheses

Solve for $x$: $5(x - 3) + 2 = 2(x + 6) - 1$

• Step 1 (Clear Fractions): No fractions present.
• Step 2 (Distribute parentheses):
  • Left side: $5 \cdot x - 5 \cdot 3 + 2 = 5x - 15 + 2$
  • Right side: $2 \cdot x + 2 \cdot 6 - 1 = 2x + 12 - 1$
• Step 3 (Combine like terms on each side independently):
  • Left side: $5x - 13$
  • Right side: $2x + 11$
  • Resulting equation: $5x - 13 = 2x + 11$
• Step 4 (Collect variable terms on the left side by subtracting $2x$):
  • $5x - 2x - 13 = 2x - 2x + 11 \implies 3x - 13 = 11$
• Step 5 (Isolate the variable):
  • Add $13$ to both sides: $3x = 11 + 13 \implies 3x = 24$
  • Divide both sides by $3$: $x = \frac{24}{3} \implies x = 8$

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Clearing Fractions Using the Least Common Denominator (LCD)

Fractions often cause calculation errors under test pressure. You can eliminate all fractional denominators in a single step by multiplying every term on both sides of the equation by the Least Common Denominator (LCD) of all fractions present.

Step-by-Step Worked Example 2: Fractional Equation

Solve for $y$: $\frac{2}{3}y - \frac{1}{2} = \frac{5}{6}$

• Step 1 (Find the LCD of 3, 2, and 6): The smallest number divisible by 3, 2, and 6 is 6.
• Step 2 (Multiply EVERY term by 6): $$6 \cdot \left(\frac{2}{3}y\right) - 6 \cdot \left(\frac{1}{2}\right) = 6 \cdot \left(\frac{5}{6}\right)$$
• Step 3 (Simplify each term):
  • $6 \cdot \frac{2}{3}y = 4y$
  • $6 \cdot \frac{1}{2} = 3$
  • $6 \cdot \frac{5}{6} = 5$
  • Cleared linear equation: $4y - 3 = 5$
• Step 4 (Solve the resulting integer equation):
  • Add $3$ to both sides: $4y = 8$
  • Divide by $4$: $y = \frac{8}{4} \implies y = 2$

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Special Cases: Conditional, Identity, and Contradiction Equations

Most linear equations on the TEAS 7 have exactly one unique solution (Conditional Equations). However, two special edge cases occasionally appear:

Example of an Identity: Solve $3(x + 2) = 3x + 6 \implies 3x + 6 = 3x + 6 \implies 6 = 6$ (True for all numbers). Example of a Contradiction: Solve $2(x - 4) = 2x + 5 \implies 2x - 8 = 2x + 5 \implies -8 = 5$ (False, No Solution).

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Healthcare Application: Medication Dosage & IV Rate Equations

Nurses frequently set up linear equations using the Desired over Have dosage formula:

$$\frac{\text{Desired Dose (D)}}{\text{Hand Dose (H)}} \times \text{Quantity (Q)} = \text{Amount to Administer (X)}$$

Clinical Problem: Oral Suspension Calculation

A physician orders $750\text{ mg}$ of Amoxicillin oral suspension for a patient with a respiratory infection. The pharmacy supplies a bottle labeled $250\text{ mg} / 5\text{ mL}$. How many milliliters ($x$) should the nurse administer?

• Step 1 (Set up the linear equation): $$\frac{750}{250} \cdot 5 = x$$
• Step 2 (Simplify the fraction ratio): $\frac{750}{250} = 3$
• Step 3 (Solve for $x$): $3 \cdot 5 = x \implies x = 15\text{ mL}$

Clinical Problem: IV Infusion Time

A patient is ordered $1,000\text{ mL}$ of $0.9\%$ Normal Saline to infuse at a constant rate of $125\text{ mL/hr}$. Set up and solve a linear equation to find the total infusion time $t$ in hours.

• Equation: $\text{Rate} \times \text{Time} = \text{Total Volume} \implies 125t = 1000$
• Solve for $t$: $t = \frac{1000}{125} = 8\text{ hours}$

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TEAS 4-Function Calculator Strategy

1. Use Scratch Paper for Isolation: Do not attempt to solve linear equations mentally or solely within the calculator. Perform inverse operations step-by-step on paper.
2. Calculator Substitution Check: After finding a solution (e.g., $x = 8$), plug it back into the original equation using the calculator to verify that Left Side = Right Side.
   • Original equation: $5(8 - 3) + 2 = 5(5) + 2 = 27$.
   • Right side: $2(8 + 6) - 1 = 2(14) - 1 = 27$. Since $27 = 27$, your solution is 100% verified.

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Common Student Traps & How to Avoid Them

1. Partial LCD Multiplication: Multiplying only the fractional terms by the LCD and forgetting to multiply whole-number constant terms. Remedy: Place brackets around the entire equation: $6 \cdot [\frac{2}{3}y - \frac{1}{2} = \frac{5}{6}]$ and distribute to every term.
2. Sign Flipping Errors when Transposing: Forgetting to change the sign of a term when moving it across the equals sign ($5x - 13 = 2x + 11 \implies 5x + 2x = 11 + 13$ is WRONG). Remedy: Write the explicit subtract operation under both sides.
3. Confusing $x = 0$ with No Solution: Obtaining $4x = 0 \implies x = 0$ and marking 'No Solution'. Remedy: Zero is a perfectly valid numerical solution! 'No Solution' only occurs when all variables cancel and leave a false statement like $0 = 7$.