6.2 Systems of Linear Equations (Elimination & Word Problems)
Key Takeaways
- The elimination (addition) method cancels one variable by multiplying one or both equations by non-zero constants to create opposite coefficients (such as $+6y$ and $-6y$), then adding the equations together.
- Substitution is most efficient when at least one variable has a coefficient of $\pm 1$; elimination is preferred when all variable coefficients are integers greater than 1 in standard form ($Ax + By = C$).
- Setting up word problems requires assigning two distinct variables with clear units, establishing two independent linear equations (typically a total quantity equation and a total value/cost equation), and solving the system.
- Common HiSET word problem archetypes include ticket sales/box office receipts, coin/currency mixtures, uniform motion with wind/water currents ($r_{\text{effective}} = r \pm c$), and percentage concentration mixtures.
- Always interpret algebraic solutions within the practical constraints of the problem—negative item counts, non-integer people, or negative speeds indicate computational errors.
Linear Systems: Elimination & Applied Word Problems
Quick Summary: The elimination method (also called the addition method) solves linear systems by adding two equations together to eliminate one variable. By multiplying one or both equations by strategic constants, we create opposite coefficients for a target variable (e.g., $+4x$ and $-4x$). On the HiSET, systems frequently appear as contextual word problems involving ticket sales, coin denominations, mixture concentrations, and headwind/tailwind motion.
While substitution is ideal when a variable is already isolated, many algebraic systems are written in standard form ($Ax + By = C$). Using elimination prevents working with messy fractions and accelerates multi-step calculations.
The Elimination Method Framework
The elimination method is founded on the Additive Property of Equality: if $A = B$ and $C = D$, then $A + C = B + D$. Adding two true equations produces another true equation.
The 5-Step Elimination Protocol
┌─────────────────────────────────────────────────────────────┐
│ Step 1: ALIGN both equations in standard form (Ax + By = C).│
│ Step 2: CHOOSE a variable to eliminate and find its LCM. │
│ Step 3: MULTIPLY one or both equations to create OPPOSITES. │
│ Step 4: ADD the equations together to eliminate the variable.│
│ Step 5: SOLVE for the remaining variable and BACK-SUBSTITUTE.│
└─────────────────────────────────────────────────────────────┘
Worked Example 1: Multiplying Both Equations
Solve the system:
Step 1: Choose a variable to eliminate. Let us eliminate $y$. The coefficients of $y$ are $5$ and $2$. The Least Common Multiple (LCM) of $5$ and $2$ is $10$.
Step 2: Create opposite coefficients ($+10y$ and $-10y$).
- Multiply Equation 1 by $2$:
- Multiply Equation 2 by $-5$:
Step 3: Add the modified equations vertically.
6x + 10y &= 2 \\ + (-35x - 10y &= -60) \\ \hline -29x + 0y &= -58 \end{array}$$ **Step 4: Solve for $x$.** $$-29x = -58 \implies x = \frac{-58}{-29} = 2$$ **Step 5: Back-substitute $x = 2$ into either original equation.** Using Equation 1: $$3(2) + 5y = 1 \implies 6 + 5y = 1$$ $$5y = 1 - 6 = -5 \implies y = -1$$ **Step 6: Check in Equation 2.** $$7(2) + 2(-1) = 14 - 2 = 12 \quad (\text{True})$$ The solution is $(2, -1)$. --- ## Decision Matrix: Substitution vs. Elimination Choosing the right method saves precious time on the HiSET exam: | System Structure | Recommended Method | Strategic Rationale | | :--- | :---: | :--- | | One equation is solved for a variable ($y = 3x - 4$) | **Substitution** | Plug expression directly into the other equation without any setup steps. | | One variable has a coefficient of $+1$ or $-1$ ($x + 4y = 9$) | **Substitution** | Isolating that variable requires only one simple subtraction/addition step. | | Both equations in standard form ($Ax + By = C$) with coefficients $> 1$ | **Elimination** | Isolating a variable produces cumbersome fractions; elimination keeps all values as integers. | | Coefficients of a variable are already opposites ($+3y$ and $-3y$) | **Elimination** | Immediate addition eliminates the variable in a single step. |HiSET Word Problem Modeling: 4 Core Archetypes
Word problems on the HiSET require translating English sentences into two algebraic equations. Below are the four most common problem structures.
Archetype 1: Total Count and Total Value (Ticket Sales / Admissions)
In this structure, one equation tracks the number of items, while the second equation tracks the monetary worth.
Worked Example: Theater Ticket Sales
A community theater sold 320 total tickets for an evening performance, generating $$3,160$ in total box office receipts. General admission tickets cost $$12$ each, while student discount tickets cost $$7$ each. How many of each ticket type were sold?
- Define the variables:
- Let $a = \text{number of adult general admission tickets}$
- Let $s = \text{number of student tickets}$
- Establish the system of equations:
- Solve via substitution ($s = 320 - a$):
- Find the student ticket count ($s$):
- Check values against total revenue: The theater sold 184 adult tickets and 136 student tickets.
Archetype 2: Coin and Currency Denomination Systems
Coin problems follow the exact same structure as ticket problems, using decimal monetary multipliers:
Pro Tip: Multiply the value equation by $100$ immediately to eliminate all decimals: $10d + 25q = 100 \times (\text{Total Value})$.
Archetype 3: Uniform Motion with Wind or Water Currents
When traveling with a tailwind or downstream current, the medium's speed adds to the vehicle's still speed. When traveling against a headwind or upstream, the medium's speed subtracts from the vehicle's speed.
Using $\text{Distance} = \text{Rate} \times \text{Time}$ ($d = rt$ or $r = \frac{d}{t}$):
Worked Example: Boat in River Current
A motorboat travels $48\text{ miles}$ downstream with the river current in $2\text{ hours}$. Returning upstream against the same current, the trip takes $3\text{ hours}$. What is the boat's speed in still water ($b$) and the speed of the river current ($c$)?
- Calculate the downstream and upstream rates:
- Downstream rate: $r_1 = \frac{48\text{ miles}}{2\text{ hours}} = 24\text{ mph}$
- Upstream rate: $r_2 = \frac{48\text{ miles}}{3\text{ hours}} = 16\text{ mph}$
- Set up the system:
- Add the equations directly (elimination of $c$):
- Find current speed $c$:
Archetype 4: Mixture and Chemical Concentration Problems
Mixture problems combine two different percentage concentrations to yield a target solution:
Where $C_1, C_2,$ and $C_{\text{final}}$ represent the decimal concentrations of pure substance.
To solve the system 3x + 4y = 18 and 5x + 6y = 28 by eliminating the variable y, which pair of multiplications produces opposite coefficients with the least common multiple?
A theater sold 250 tickets for a weekend performance, generating $2,100 in total revenue. Adult tickets cost $10 each and student tickets cost $6 each. How many adult tickets were sold?
A small aircraft flies 600 miles with a tailwind in 3 hours. On the return trip against the same headwind, the flight takes 4 hours. What is the airspeed of the plane in still air?
A chemist needs to combine a 20% acid solution with a 50% acid solution to create 60 liters of a 30% acid solution. How many liters of the 20% solution and 50% solution must be mixed?