Linear Equations in Two Variables Pattern - Word Problems

What This Pattern Is About

These questions give you a real-world situation involving two categories (items, activities, ingredients, etc.) and ask you to either write the correct linear equation that models it, or substitute a given value into an equation and solve for the unknown. This is the most heavily tested pattern in the two-variable linear equations skill — almost every question boils down to one of two tasks: "which equation?" or "plug in and solve."

 

The Core Idea

Most word problems follow the same template:

$$\text{(rate}_1\text{)(quantity}_1\text{)} + \text{(rate}_2\text{)(quantity}_2\text{)} = \text{total}$$

Your job is to match each rate to the correct variable and make sure the right-hand side represents the overall total. Once the equation is set up, solving is straightforward algebra.

 

Sub-Type 1 — Choose the Correct Equation

The question describes a scenario and gives you four candidate equations. The most common trap is swapping the coefficients — attaching the wrong rate to the wrong variable.

A school club is selling cupcakes for $2 each and pies for $5 each. The club's goal is to raise a total of $350. Which equation represents this situation, where $x$ is the number of cupcakes sold and $y$ is the number of pies sold?

(A) $5x + 2y = 350$   (B) $2x + 5y = 350$   (C) $7(x + y) = 350$   (D) $7xy = 350$

Identify the rates. Cupcakes cost $2 each, and $x$ counts cupcakes, so the cupcake revenue is $2x$. Pies cost $5 each, and $y$ counts pies, so the pie revenue is $5y$.

Build the equation. Total revenue = cupcake revenue + pie revenue: $2x + 5y = 350$.

Check the traps. (A) swaps the prices. (C) assumes every item costs $7. (D) multiplies instead of adding. Answer: B

 

Sub-Type 2 — Substitute and Solve

The question provides a linear equation and gives you one variable's value. You substitute that value and solve for the other.

A chemist is using 50-mL beakers and 100-mL beakers to measure a total of 1,650 mL of solution. The chemist used exactly 12 of the 100-mL beakers. How many 50-mL beakers were used?

(A) 9   (B) 12   (C) 50   (D) 100

Set up the equation. Let $x$ be the number of 50-mL beakers and $y$ be the number of 100-mL beakers: $50x + 100y = 1{,}650$.

Substitute $y = 12$. $50x + 100(12) = 1{,}650$, so $50x + 1{,}200 = 1{,}650$.

Solve. $50x = 450$, giving $x = 9$. Answer: A

 

The equation $15x + 5y = 150$ represents the budget for a catering order, where $x$ is the number of main courses and $y$ is the number of side dishes. If a client orders 6 main courses, how many side dishes can they order?

(A) 8   (B) 12   (C) 15   (D) 30

Substitute $x = 6$. $15(6) + 5y = 150$, so $90 + 5y = 150$.

Solve. $5y = 60$, giving $y = 12$. Answer: B

 

Sub-Type 3 — Write an Equation from Slope and a Point

Some questions give a slope and a point (usually the $y$-intercept) and ask for the line's equation. Just plug into $y = mx + b$.

A line in the $xy$-plane has a slope of $-\dfrac{2}{3}$ and passes through the point $(0, 4)$. Which equation represents this line?

(A) $y = \dfrac{2}{3}x + 4$   (B) $y = -\dfrac{2}{3}x - 4$   (C) $y = \dfrac{2}{3}x - 4$   (D) $y = -\dfrac{2}{3}x + 4$

Identify $m$ and $b$. The slope is $m = -\dfrac{2}{3}$. The point $(0, 4)$ is on the $y$-axis, so $b = 4$.

Plug in. $y = -\dfrac{2}{3}x + 4$. Answer: D

 

Sub-Type 4 — Mixture / Percent Concentration Setup

These questions involve mixing two solutions with different concentrations. The equation tracks the total amount of pure substance (not total volume).

A scientist mixes $r$ liters of rainwater (1% pollutant) with $s$ liters of stream water (18% pollutant). The result is 10 liters of a 5% pollutant mixture. Which equation describes this?

(A) $0.18r + 0.01s = (0.05)(10)$   (B) $0.01r + 0.18s = 10$   (C) $0.01r + 0.18s = (0.05)(10)$   (D) $0.18r + 0.01s = 10$

Track the pollutant, not the volume. Rainwater contributes $0.01r$ liters of pollutant. Stream water contributes $0.18s$ liters. The final mixture has $(0.05)(10) = 0.5$ liters of pollutant.

Build the equation. $0.01r + 0.18s = (0.05)(10)$.

Check the traps. (A) swaps the concentrations. (B) and (D) set the right side equal to the total volume (10) instead of the total pollutant. Answer: C

 

Sub-Type 5 — Percent Increase Setup

These questions describe a percent increase in two categories and an overall percent increase. You model the new amounts using multipliers like $1.22x$ (a 22% increase).

An orchard produced 800 tons of fruit last year (apples and pears). This year, apple production rose 22%, pear production rose 35%, and total production rose 27%. Which equation models this, where $x$ is last year's apple production and $y$ is last year's pear production?

(A) $1.35x + 1.22y = 800(1.27)$   (B) $1.22x + 1.35y = 800(1.27)$   (C) $1.22x + 1.27y = 800(1.35)$   (D) $1.27x + 1.35y = 800(1.22)$

Convert each increase to a multiplier. Apples: $x \to 1.22x$. Pears: $y \to 1.35y$. Total: $800 \to 800(1.27)$.

Build the equation. New apples + new pears = new total: $1.22x + 1.35y = 800(1.27)$.

Check the traps. (A) swaps the growth rates for the two fruits. Answer: B

 

Sub-Type 6 — "N Times as Many" with Substitution

Some problems give two relationships: a cost/total equation and a ratio between the variables. Substitute the ratio into the main equation to get a single variable.

A workout burns 110 calories per mile of running and 40 calories per mile of cycling. A person burned 1,240 total calories and cycled five times as many miles as they ran. How many miles did the person run?

(A) 5   (B) 4   (C) 24   (D) 20

Define variables. Let $r$ = miles run, $c$ = miles cycled. Cost equation: $110r + 40c = 1{,}240$. Ratio: $c = 5r$.

Substitute. $110r + 40(5r) = 1{,}240$, so $110r + 200r = 1{,}240$, giving $310r = 1{,}240$.

Solve. $r = 4$. Answer: B

 

Sub-Type 7 — Multi-Step Coefficient Computation

At the Medium level, the coefficient itself may require a multiplication before you can write the equation. Read carefully to combine all the given numbers.

A fitness center buys 3 sets of dumbbells and 5 sets of kettlebells. Each dumbbell set has 10 dumbbells weighing $d$ pounds each; each kettlebell set has 6 kettlebells weighing $k$ pounds each. The total weight is 1,200 pounds. Which equation describes this?

(A) $3d + 5k = 1{,}200$   (B) $50d + 18k = 1{,}200$   (C) $18d + 50k = 1{,}200$   (D) $30d + 30k = 1{,}200$

Compute each coefficient. Total dumbbells: $3 \times 10 = 30$, so total dumbbell weight = $30d$. Total kettlebells: $5 \times 6 = 30$, so total kettlebell weight = $30k$.

Build the equation. $30d + 30k = 1{,}200$.

Check the traps. (A) ignores the items-per-set step. (B) and (C) swap the set counts. Answer: D

 

Sub-Type 8 — Three-Step: Total, Subtract, Divide

These questions give a total, a known sub-total, and a count, and ask for a per-unit value. The recipe is always: compute the known portion, subtract from the total, then divide by the count.

A plumber uses 486 inches of pipe: 6 long pipes (48 inches each) and 11 short pipes. What is the length of one short pipe?

Step 1 — Compute the known portion. $6 \times 48 = 288$ inches of long pipe.

Step 2 — Subtract. $486 - 288 = 198$ inches of short pipe total.

Step 3 — Divide. $\dfrac{198}{11} = 18$ inches per short pipe. Answer: 18

 

How to Recognize This Pattern

Look for a word problem that either asks "which equation represents this situation?" or gives you an equation and one value and asks you to find the other. The context always involves two categories contributing to a single total.

Common Mistakes to Avoid

• Swapping coefficients: the most frequent trap answer. Always double-check that each rate is multiplied by its matching variable.
• Confusing count with total: if you are asked "how many items?" but pick the answer that represents total cost or total weight, you have mixed up the question.
• Stopping one step short: on the three-step problems, students often find the sub-total (e.g., 198 inches of short pipe) and pick that instead of dividing by the count.
• On mixture problems, setting the right side equal to the total volume instead of the total amount of pure substance.
• On percent-increase problems, using the raw percentages (0.22) instead of the multipliers (1.22).