How to Solve Systems of Equations Word Problems

System of Equations Word Problem?

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How to Model a Word Problem

A system of equations word problem gives two or more facts about the same unknown quantities. Translate each fact into an equation, solve the system, and then answer the question in the language of the problem.

  • Define the variables before writing equations.
  • Use one equation for each relationship.
  • Solve by substitution or elimination.
  • Check that the values fit the context and units.

Example

A theater sold adult tickets for \(x\) dollars and child tickets for \(y\) dollars. Two adult tickets and three child tickets cost $39. Four adult tickets and one child ticket cost $51.

  • Write the system: \(2x + 3y = 39\) and \(4x + y = 51\).
  • Multiply the second equation by \(3\): \(12x + 3y = 153\).
  • Subtract the first equation: \(10x = 114\), so \(x = 11.4\).
  • Substitute back: \(4(11.4) + y = 51\), so \(y = 5.4\).

The adult ticket costs $11.40, and the child ticket costs $5.40.

Systems of Equations Word Problems

Think of this lesson as more than a rule to memorize. Systems of Equations Word Problems is about undoing operations while keeping both sides balanced. A strong student does not rush to the first formula on the page; they pause, identify the structure of the problem, and then choose the tool that matches that structure. That pause is what prevents most mistakes.

An equation is a balance. Whatever operation you use on one side, you must use on the other side so the two expressions stay equal.

Here is the teacher way to approach the topic. First, read the problem slowly and underline the information that is actually given. Next, name the target: are you finding a value, simplifying an expression, comparing two quantities, solving for a variable, or interpreting a graph? Once the target is clear, the calculation becomes much less mysterious because every step has a job.

  • Clear clutter such as parentheses or fractions.
  • Collect like terms.
  • Undo operations in reverse order.
  • Substitute the answer back or test a point.

A helpful habit is to say what each number represents before using it. For example, if a number is a denominator, a radius, a slope, a common difference, or a coefficient, it should not be treated like an ordinary loose number. Its role tells you where it belongs in the formula. This is especially important on ACT-style questions because many wrong answer choices come from using the right numbers in the wrong places.

Another good habit is to keep the work organized vertically. Write one clean line for substitution, one line for simplifying, and one line for the final answer. If the problem has units, keep the units attached. If the problem has signs, exponents, or parentheses, copy them carefully from one line to the next. Most errors in this topic are not caused by a hard idea; they are caused by dropping a negative sign, combining unlike terms, using the wrong denominator, or skipping a check.

When you finish, ask a quick reasonableness question. Should the answer be positive or negative? Should it be larger or smaller than the original number? Does it fit the graph, table, shape, or equation? Can you plug it back into the original problem? This final check turns the lesson from a procedure into understanding.

On a test, the goal is not to write the longest solution. The goal is to write enough clear work that you can see the structure, avoid traps, and recover quickly if one line goes wrong. Practice the examples below with that mindset: identify the type of problem, choose the matching rule, show the substitution, simplify carefully, and check the answer before moving on.

Free printable Worksheets

Exercises for Systems of Equations Word Problems

1) Adult tickets cost \(x\) dollars and student tickets cost \(y\) dollars. Three adult tickets and two student tickets cost $44. Two adult tickets and four student tickets cost $40. Find both ticket prices.
System: \( \begin{cases} 3x + 2y = 44 \\ 2x + 4y = 40 \end{cases} \)

2) A snack stand sells pretzels for \(x\) dollars and drinks for \(y\) dollars. Four pretzels and three drinks cost $25. Two pretzels and five drinks cost $23. Find each price.
System: \( \begin{cases} 4x + 3y = 25 \\ 2x + 5y = 23 \end{cases} \)

3) A farm has chickens and goats. There are \(18\) animals and \(50\) legs total. Let \(x\) be chickens and \(y\) be goats. How many of each are there?
System: \( \begin{cases} x + y = 18 \\ 2x + 4y = 50 \end{cases} \)

4) A club sold small shirts for \(x\) dollars and large shirts for \(y\) dollars. Five small and three large shirts cost $102. Two small and six large shirts cost $108. Find both prices.
System: \( \begin{cases} 5x + 3y = 102 \\ 2x + 6y = 108 \end{cases} \)

5) Two numbers have a sum of \(31\). Three times the first number plus two times the second number is \(82\). Find the numbers.
System: \( \begin{cases} x + y = 31 \\ 3x + 2y = 82 \end{cases} \)

6) A parking lot has cars and motorcycles. There are \(42\) vehicles and \(128\) wheels. Let \(x\) be cars and \(y\) be motorcycles. How many are there of each?
System: \( \begin{cases} x + y = 42 \\ 4x + 2y = 128 \end{cases} \)

7) A bakery sells muffins for \(x\) dollars and bagels for \(y\) dollars. Six muffins and four bagels cost $30. Three muffins and eight bagels cost $33. Find each price.
System: \( \begin{cases} 6x + 4y = 30 \\ 3x + 8y = 33 \end{cases} \)

8) A taxi company charges a base fee of \(x\) dollars plus \(y\) dollars per mile. A \(5\)-mile ride costs $18. A \(9\)-mile ride costs $30. Find the base fee and mileage rate.
System: \( \begin{cases} x + 5y = 18 \\ x + 9y = 30 \end{cases} \)

9) A school bought calculators for \(x\) dollars each and notebooks for \(y\) dollars each. Seven calculators and ten notebooks cost $155. Four calculators and fifteen notebooks cost $135. Find both prices.
System: \( \begin{cases} 7x + 10y = 155 \\ 4x + 15y = 135 \end{cases} \)

10) A jar contains nickels and dimes. There are \(46\) coins worth \(340\) cents. Let \(x\) be nickels and \(y\) be dimes. How many of each coin are there?
System: \( \begin{cases} x + y = 46 \\ 5x + 10y = 340 \end{cases} \)

11) The perimeter of a rectangle is \(58\) inches. Its length is \(5\) inches more than its width. Let \(x\) be the length and \(y\) be the width. Find both dimensions.
System: \( \begin{cases} 2x + 2y = 58 \\ x - y = 5 \end{cases} \)

12) A concert sold balcony tickets for \(x\) dollars and floor tickets for \(y\) dollars. Twelve balcony and eight floor tickets cost $680. Five balcony and ten floor tickets cost $550. Find each ticket price.
System: \( \begin{cases} 12x + 8y = 680 \\ 5x + 10y = 550 \end{cases} \)

13) Two investments total $900. One earns \(4\%\) interest and the other earns \(7\%\). The total annual interest is $51. Let \(x\) and \(y\) be the amounts invested. Find both amounts.
System: \( \begin{cases} x + y = 900 \\ 4x + 7y = 5100 \end{cases} \)

14) A boat travels downstream at \(18\) mph and upstream at \(8\) mph. Let \(x\) be the boat speed in still water and \(y\) be the current speed. Find both speeds.
System: \( \begin{cases} x + y = 18 \\ x - y = 8 \end{cases} \)

15) A chemist mixes \(x\) liters of \(20\%\) solution and \(y\) liters of \(50\%\) solution to make \(30\) liters of \(38\%\) solution. Find \(x\) and \(y\).
System: \( \begin{cases} x + y = 30 \\ 20x + 50y = 1140 \end{cases} \)

16) A theater group sold \(85\) tickets. Regular tickets cost $18 and premium tickets cost $30. Total sales were $1,830. How many of each ticket were sold?
System: \( \begin{cases} x + y = 85 \\ 18x + 30y = 1830 \end{cases} \)

17) A test has \(25\) questions worth either \(2\) points or \(5\) points. The test is worth \(83\) points total. How many questions of each type are there?
System: \( \begin{cases} x + y = 25 \\ 2x + 5y = 83 \end{cases} \)

18) Two printers work together. In \(6\) minutes together they print \(570\) pages; in \(4\) minutes Printer A and \(3\) minutes Printer B print \(320\) pages. Find both rates.
System: \( \begin{cases} 6x + 6y = 570 \\ 4x + 3y = 320 \end{cases} \)

19) A store sells two laptop models. Four basic laptops and three advanced laptops cost $4,700. Two basic laptops and five advanced laptops cost $5,500. Find each price.
System: \( \begin{cases} 4x + 3y = 4700 \\ 2x + 5y = 5500 \end{cases} \)

20) A rowing team travels downstream at \(21\) mph and upstream at \(13\) mph. Let \(x\) be still-water speed and \(y\) be current speed. Find both speeds.
System: \( \begin{cases} x + y = 21 \\ x - y = 13 \end{cases} \)

 
1)

Let \(x\) be the adult ticket price and \(y\) be the student ticket price.

Start with \(3x + 2y = 44\) and \(2x + 4y = 40\).

Eliminate \(y\): multiply the first equation by \(4\) and the second equation by \(2\), then subtract.

\((3)(4)x - (2)(2)x = (44)(4) - (40)(2)\), so \(8x = 96\).

Divide to get \(x = 12\).

Substitute into the first equation: \(3(12) + 2y = 44\), so \(36 + 2y = 44\) and \(y = 4\).

Answer: \((x, y) = (12, 4)\).

Adult tickets cost $12, and student tickets cost $4.

2)

Let \(x\) be the pretzel price and \(y\) be the drink price.

Start with \(4x + 3y = 25\) and \(2x + 5y = 23\).

Eliminate \(y\): multiply the first equation by \(5\) and the second equation by \(3\), then subtract.

\((4)(5)x - (2)(3)x = (25)(5) - (23)(3)\), so \(14x = 56\).

Divide to get \(x = 4\).

Substitute into the first equation: \(4(4) + 3y = 25\), so \(16 + 3y = 25\) and \(y = 3\).

Answer: \((x, y) = (4, 3)\).

A pretzel costs $4, and a drink costs $3.

3)

Let \(x\) be chickens and \(y\) be goats.

Start with \(x + y = 18\) and \(2x + 4y = 50\).

Eliminate \(y\): multiply the first equation by \(4\) and the second equation by \(1\), then subtract.

\((1)(4)x - (2)(1)x = (18)(4) - (50)(1)\), so \(2x = 22\).

Divide to get \(x = 11\).

Substitute into the first equation: \(1(11) + 1y = 18\), so \(11 + 1y = 18\) and \(y = 7\).

Answer: \((x, y) = (11, 7)\).

There are \(11\) chickens and \(7\) goats.

4)

Let \(x\) be the small-shirt price and \(y\) be the large-shirt price.

Start with \(5x + 3y = 102\) and \(2x + 6y = 108\).

Eliminate \(y\): multiply the first equation by \(6\) and the second equation by \(3\), then subtract.

\((5)(6)x - (2)(3)x = (102)(6) - (108)(3)\), so \(24x = 288\).

Divide to get \(x = 12\).

Substitute into the first equation: \(5(12) + 3y = 102\), so \(60 + 3y = 102\) and \(y = 14\).

Answer: \((x, y) = (12, 14)\).

A small shirt costs $12, and a large shirt costs $14.

5)

Let \(x\) be the first number and \(y\) be the second number.

Start with \(x + y = 31\) and \(3x + 2y = 82\).

Eliminate \(y\): multiply the first equation by \(2\) and the second equation by \(1\), then subtract.

\((1)(2)x - (3)(1)x = (31)(2) - (82)(1)\), so \(-1x = -20\).

Divide to get \(x = 20\).

Substitute into the first equation: \(1(20) + 1y = 31\), so \(20 + 1y = 31\) and \(y = 11\).

Answer: \((x, y) = (20, 11)\).

The numbers are \(20\) and \(11\).

6)

Let \(x\) be cars and \(y\) be motorcycles.

Start with \(x + y = 42\) and \(4x + 2y = 128\).

Eliminate \(y\): multiply the first equation by \(2\) and the second equation by \(1\), then subtract.

\((1)(2)x - (4)(1)x = (42)(2) - (128)(1)\), so \(-2x = -44\).

Divide to get \(x = 22\).

Substitute into the first equation: \(1(22) + 1y = 42\), so \(22 + 1y = 42\) and \(y = 20\).

Answer: \((x, y) = (22, 20)\).

There are \(22\) cars and \(20\) motorcycles.

7)

Let \(x\) be the muffin price and \(y\) be the bagel price.

Start with \(6x + 4y = 30\) and \(3x + 8y = 33\).

Eliminate \(y\): multiply the first equation by \(8\) and the second equation by \(4\), then subtract.

\((6)(8)x - (3)(4)x = (30)(8) - (33)(4)\), so \(36x = 108\).

Divide to get \(x = 3\).

Substitute into the first equation: \(6(3) + 4y = 30\), so \(18 + 4y = 30\) and \(y = 3\).

Answer: \((x, y) = (3, 3)\).

A muffin costs $3, and a bagel costs $3.

8)

Let \(x\) be the base fee and \(y\) be the cost per mile.

Start with \(x + 5y = 18\) and \(x + 9y = 30\).

Eliminate \(y\): multiply the first equation by \(9\) and the second equation by \(5\), then subtract.

\((1)(9)x - (1)(5)x = (18)(9) - (30)(5)\), so \(4x = 12\).

Divide to get \(x = 3\).

Substitute into the first equation: \(1(3) + 5y = 18\), so \(3 + 5y = 18\) and \(y = 3\).

Answer: \((x, y) = (3, 3)\).

The base fee is $3, and the rate is $3 per mile.

9)

Let \(x\) be the calculator price and \(y\) be the notebook price.

Start with \(7x + 10y = 155\) and \(4x + 15y = 135\).

Eliminate \(y\): multiply the first equation by \(15\) and the second equation by \(10\), then subtract.

\((7)(15)x - (4)(10)x = (155)(15) - (135)(10)\), so \(65x = 975\).

Divide to get \(x = 15\).

Substitute into the first equation: \(7(15) + 10y = 155\), so \(105 + 10y = 155\) and \(y = 5\).

Answer: \((x, y) = (15, 5)\).

A calculator costs $15, and a notebook costs $5.

10)

Let \(x\) be nickels and \(y\) be dimes.

Start with \(x + y = 46\) and \(5x + 10y = 340\).

Eliminate \(y\): multiply the first equation by \(10\) and the second equation by \(1\), then subtract.

\((1)(10)x - (5)(1)x = (46)(10) - (340)(1)\), so \(5x = 120\).

Divide to get \(x = 24\).

Substitute into the first equation: \(1(24) + 1y = 46\), so \(24 + 1y = 46\) and \(y = 22\).

Answer: \((x, y) = (24, 22)\).

There are \(24\) nickels and \(22\) dimes.

11)

Let \(x\) be length and \(y\) be width.

Start with \(2x + 2y = 58\) and \(x - y = 5\).

Eliminate \(y\): multiply the first equation by \(-1\) and the second equation by \(2\), then subtract.

\((2)(-1)x - (1)(2)x = (58)(-1) - (5)(2)\), so \(-4x = -68\).

Divide to get \(x = 17\).

Substitute into the first equation: \(2(17) + 2y = 58\), so \(34 + 2y = 58\) and \(y = 12\).

Answer: \((x, y) = (17, 12)\).

The length is \(17\) inches, and the width is \(12\) inches.

12)

Let \(x\) be the balcony price and \(y\) be the floor price.

Start with \(12x + 8y = 680\) and \(5x + 10y = 550\).

Eliminate \(y\): multiply the first equation by \(10\) and the second equation by \(8\), then subtract.

\((12)(10)x - (5)(8)x = (680)(10) - (550)(8)\), so \(80x = 2400\).

Divide to get \(x = 30\).

Substitute into the first equation: \(12(30) + 8y = 680\), so \(360 + 8y = 680\) and \(y = 40\).

Answer: \((x, y) = (30, 40)\).

A balcony ticket costs $30, and a floor ticket costs $40.

13)

Let \(x\) be the amount at \(4\%\) and \(y\) be the amount at \(7\%\).

Start with \(x + y = 900\) and \(4x + 7y = 5100\).

Eliminate \(y\): multiply the first equation by \(7\) and the second equation by \(1\), then subtract.

\((1)(7)x - (4)(1)x = (900)(7) - (5100)(1)\), so \(3x = 1200\).

Divide to get \(x = 400\).

Substitute into the first equation: \(1(400) + 1y = 900\), so \(400 + 1y = 900\) and \(y = 500\).

Answer: \((x, y) = (400, 500)\).

$400 is invested at \(4\%\), and $500 is invested at \(7\%\).

14)

Let \(x\) be boat speed and \(y\) be current speed.

Start with \(x + y = 18\) and \(x - y = 8\).

Eliminate \(y\): multiply the first equation by \(-1\) and the second equation by \(1\), then subtract.

\((1)(-1)x - (1)(1)x = (18)(-1) - (8)(1)\), so \(-2x = -26\).

Divide to get \(x = 13\).

Substitute into the first equation: \(1(13) + 1y = 18\), so \(13 + 1y = 18\) and \(y = 5\).

Answer: \((x, y) = (13, 5)\).

The boat speed is \(13\) mph, and the current speed is \(5\) mph.

15)

Let \(x\) be liters of \(20\%\) solution and \(y\) be liters of \(50\%\) solution.

Start with \(x + y = 30\) and \(20x + 50y = 1140\).

Eliminate \(y\): multiply the first equation by \(50\) and the second equation by \(1\), then subtract.

\((1)(50)x - (20)(1)x = (30)(50) - (1140)(1)\), so \(30x = 360\).

Divide to get \(x = 12\).

Substitute into the first equation: \(1(12) + 1y = 30\), so \(12 + 1y = 30\) and \(y = 18\).

Answer: \((x, y) = (12, 18)\).

Use \(12\) liters of \(20\%\) solution and \(18\) liters of \(50\%\) solution.

16)

Let \(x\) be regular tickets and \(y\) be premium tickets.

Start with \(x + y = 85\) and \(18x + 30y = 1830\).

Eliminate \(y\): multiply the first equation by \(30\) and the second equation by \(1\), then subtract.

\((1)(30)x - (18)(1)x = (85)(30) - (1830)(1)\), so \(12x = 720\).

Divide to get \(x = 60\).

Substitute into the first equation: \(1(60) + 1y = 85\), so \(60 + 1y = 85\) and \(y = 25\).

Answer: \((x, y) = (60, 25)\).

They sold \(60\) regular tickets and \(25\) premium tickets.

17)

Let \(x\) be \(2\)-point questions and \(y\) be \(5\)-point questions.

Start with \(x + y = 25\) and \(2x + 5y = 83\).

Eliminate \(y\): multiply the first equation by \(5\) and the second equation by \(1\), then subtract.

\((1)(5)x - (2)(1)x = (25)(5) - (83)(1)\), so \(3x = 42\).

Divide to get \(x = 14\).

Substitute into the first equation: \(1(14) + 1y = 25\), so \(14 + 1y = 25\) and \(y = 11\).

Answer: \((x, y) = (14, 11)\).

There are \(14\) two-point questions and \(11\) five-point questions.

18)

Let \(x\) be Printer A rate and \(y\) be Printer B rate.

Start with \(6x + 6y = 570\) and \(4x + 3y = 320\).

Eliminate \(y\): multiply the first equation by \(3\) and the second equation by \(6\), then subtract.

\((6)(3)x - (4)(6)x = (570)(3) - (320)(6)\), so \(-6x = -210\).

Divide to get \(x = 35\).

Substitute into the first equation: \(6(35) + 6y = 570\), so \(210 + 6y = 570\) and \(y = 60\).

Answer: \((x, y) = (35, 60)\).

Printer A prints \(35\) pages per minute, and Printer B prints \(60\) pages per minute.

19)

Let \(x\) be the basic laptop price and \(y\) be the advanced laptop price.

Start with \(4x + 3y = 4700\) and \(2x + 5y = 5500\).

Eliminate \(y\): multiply the first equation by \(5\) and the second equation by \(3\), then subtract.

\((4)(5)x - (2)(3)x = (4700)(5) - (5500)(3)\), so \(14x = 7000\).

Divide to get \(x = 500\).

Substitute into the first equation: \(4(500) + 3y = 4700\), so \(2000 + 3y = 4700\) and \(y = 900\).

Answer: \((x, y) = (500, 900)\).

A basic laptop costs $500, and an advanced laptop costs $900.

20)

Let \(x\) be still-water speed and \(y\) be current speed.

Start with \(x + y = 21\) and \(x - y = 13\).

Eliminate \(y\): multiply the first equation by \(-1\) and the second equation by \(1\), then subtract.

\((1)(-1)x - (1)(1)x = (21)(-1) - (13)(1)\), so \(-2x = -34\).

Divide to get \(x = 17\).

Substitute into the first equation: \(1(17) + 1y = 21\), so \(17 + 1y = 21\) and \(y = 4\).

Answer: \((x, y) = (17, 4)\).

The still-water speed is \(17\) mph, and the current speed is \(4\) mph.

Systems of Equations Word Problems Quiz