Find Points for Silver and Gold Coins

Question

PROBLEM

Bernie and Kenny are playing Bernie's new video game, in which players earn points by finding coins. In the first game, Bernie finds 4 silver coins and 2 gold coins for a total of 28 points. Kenny finds 3 silver coins and 3 gold coins for a total of 30 points. How many points do the players earn by finding a silver coin? How many points do they earn by finding a gold coin?

What This Problem Teaches

Setting up a system of linear equations from word problem constraints
Recognizing when two unknowns require two separate equations
Solving systems using elimination and substitution methods
Translating between mathematical notation and real-world scenarios
Verifying solutions by checking against original conditions

Visualizing the Game

Bernie and Kenny are playing Bernie's new video game, in which players earn points by finding coins. In the first...

Solution: Method 1 — The System of Equations Approach

Step 1 — Define variables for the unknown point values

Let s = points earned by finding one silver coin

Let g = points earned by finding one gold coin

Step 2 — Translate Bernie's game into an equation

Bernie finds 4 silver coins and 2 gold coins for 28 points total. This gives us:

4s + 2g = 28

Step 3 — Translate Kenny's game into a second equation

Kenny finds 3 silver coins and 3 gold coins for 30 points total. This gives us:

3s + 3g = 30

Step 4 — Simplify the second equation

We can divide Kenny's equation by 3 to make the coefficients smaller:

3s + 3g = 30
s + g = 10

Step 5 — Solve using substitution

From the simplified equation, we can express g in terms of s:

s + g = 10
g = 10 - s

Substitute this into Bernie's equation:

4s + 2g = 28
4s + 2(10 - s) = 28
4s + 20 - 2s = 28
2s + 20 = 28
2s = 8
s = 4

Step 6 — Find the value of gold coins

Now substitute s = 4 back into g = 10 - s:

g = 10 - 4 = 6

Solution: Method 2 — The Elimination Strategy

Step 1 — Set up the system of equations

Starting with our two equations from the problem:

4s + 2g = 28 ... (Equation 1)
3s + 3g = 30 ... (Equation 2)

Step 2 — Prepare equations for elimination

To eliminate one variable, we need matching coefficients. Let's eliminate s by making the coefficients of s the same. Multiply Equation 1 by 3 and Equation 2 by 4:

3(4s + 2g) = 3(28) → 12s + 6g = 84
4(3s + 3g) = 4(30) → 12s + 12g = 120

Step 3 — Eliminate s by subtraction

Subtract the first equation from the second:

(12s + 12g) - (12s + 6g) = 120 - 84
12s + 12g - 12s - 6g = 36
6g = 36
g = 6

Step 4 — Substitute back to find s

Use g = 6 in the original first equation:

4s + 2(6) = 28
4s + 12 = 28
4s = 16
s = 4

Silver coins are worth 4 points each, and gold coins are worth 6 points each.

Verification

Let's check our answer by substituting s = 4 and g = 6 back into both original conditions:

Bernie's game check:

4 silver + 2 gold = 4(4) + 2(6) = 16 + 12 = 28 ✓

Kenny's game check:

3 silver + 3 gold = 3(4) + 3(6) = 12 + 18 = 30 ✓

Both totals match perfectly, confirming our solution is correct.

Common Pitfalls

✗ Assuming gold and silver are worth the same

Some students try s = g and solve 6s = 28 from Bernie's game, getting s = 4.67. But then Kenny's game gives 6s = 30, so s = 5. The contradiction shows the coins must have different values.

✗ Setting up only one equation

With two unknowns, you need two independent equations. Using only Bernie's data gives infinitely many solutions — silver could be 2 points and gold 10, or silver could be 6 points and gold 2, etc.

✗ Mixing up the coefficients

Writing 2s + 4g = 28 instead of 4s + 2g = 28 flips which coin Bernie found more of. Always double-check that your equation matches the problem description.

The General Formula

Any problem where you have two unknowns and two linear constraints follows this pattern:

a₁x + b₁y = c₁
a₂x + b₂y = c₂

The solution exists and is unique when the equations are not multiples of each other (mathematically, when the determinant a₁b₂ - a₂b₁ ≠ 0). In our coin problem, this determinant is (4)(3) - (3)(2) = 12 - 6 = 6 ≠ 0, confirming a unique solution exists.

Real-world insight: This same mathematical structure appears everywhere — mixing concentrations in chemistry, balancing nutrients in diet planning, or determining hourly rates for different types of work.

Four "What-If?" Problems

1

Different Point Values

Bernie finds 3 silver coins and 4 gold coins for a total of 39 points. Kenny finds 2 silver coins and 1 gold coin for a total of 16 points. How many points are silver and gold coins worth?

Step 1 — Set up the equations

Let s = silver points, g = gold points. Bernie: 3s + 4g = 39. Kenny: 2s + 1g = 16.

Step 2 — Solve for g from Kenny's equation

2s + g = 16 → g = 16 - 2s

Step 3 — Substitute into Bernie's equation

3s + 4(16 - 2s) = 39 → 3s + 64 - 8s = 39 → -5s = -25 → s = 5

Step 4 — Find g

g = 16 - 2(5) = 6

Step 5 — Verify

Bernie: 3(5) + 4(6) = 15 + 24 = 39 ✓. Kenny: 2(5) + 1(6) = 16 ✓

Silver = 5 points, Gold = 6 points

2

Adding Bronze Coins

The game now has three coin types. Bernie finds 2 silver, 1 gold, and 3 bronze for 25 points. Kenny finds 1 silver, 2 gold, and 1 bronze for 23 points. If bronze coins are worth 3 points each, what are silver and gold worth?

Step 1 — Account for bronze points

Bronze = 3 points each. Bernie's bronze contribution: 3 × 3 = 9. Kenny's: 1 × 3 = 3.

Step 2 — Set up equations for silver and gold only

Bernie: 2s + 1g + 9 = 25 → 2s + g = 16

Kenny: 1s + 2g + 3 = 23 → s + 2g = 20

Step 3 — Solve the 2×2 system

From equation 1: g = 16 - 2s

Substitute: s + 2(16 - 2s) = 20 → s + 32 - 4s = 20 → -3s = -12 → s = 4

Step 4 — Find g

g = 16 - 2(4) = 8

Step 5 — Verify

Bernie: 2(4) + 1(8) + 3(3) = 8 + 8 + 9 = 25 ✓

Kenny: 1(4) + 2(8) + 1(3) = 4 + 16 + 3 = 23 ✓

Silver = 4 points, Gold = 8 points

3

Reverse the Unknown

Bernie finds 5 silver and 1 gold for 26 points. Kenny finds 2 silver and 4 gold for 32 points. If you know that silver coins are worth 4 points each, how many points is a gold coin worth?

Step 1 — Use Bernie's game to find gold value

Bernie: 5s + 1g = 26, where s = 4

5(4) + g = 26 → 20 + g = 26 → g = 6

Step 2 — Verify with Kenny's game

Kenny: 2s + 4g = 2(4) + 4(6) = 8 + 24 = 32 ✓

Step 3 — Alternative approach

We could also use Kenny's equation: 2(4) + 4g = 32 → 8 + 4g = 32 → 4g = 24 → g = 6

Final Answer

Gold coins are worth 6 points each.

4

Third Player Challenge

Using the original point values (silver = 4, gold = 6), a new player Lisa finds 6 silver coins and 2 gold coins. Another player Mike gets exactly 40 points with some combination of silver and gold coins. Give three different combinations of coins Mike could have found.

Step 1 — Find Lisa's score

Lisa: 6 × 4 + 2 × 6 = 24 + 12 = 36 points

Step 2 — Set up equation for Mike

Mike needs: 4s + 6g = 40, where s and g are non-negative integers

Step 3 — Find valid combinations

Rearrange: s = (40 - 6g)/4 = 10 - 1.5g

For s to be a whole number, g must be even.

Try even values: g = 0 → s = 10; g = 2 → s = 7; g = 4 → s = 4; g = 6 → s = 1

Step 4 — Verify three solutions

Option 1: 10 silver, 0 gold → 10(4) + 0(6) = 40 ✓

Option 2: 7 silver, 2 gold → 7(4) + 2(6) = 28 + 12 = 40 ✓

Option 3: 4 silver, 4 gold → 4(4) + 4(6) = 16 + 24 = 40 ✓

Frequently Asked Questions

How do you solve a system of equations with two unknowns?+

Set up one equation for each constraint, then use substitution or elimination. In this video game problem, Bernie's coins give us 4s + 2g = 28, Kenny's give us 3s + 3g = 30. Solving these simultaneously reveals s = 4 points and g = 6 points.

What's the difference between substitution and elimination methods?+

Substitution solves one equation for a variable, then plugs that expression into the other equation. Elimination adds or subtracts equations to cancel out one variable. Both work - elimination is often faster when coefficients line up nicely, like in this coin problem where we can easily eliminate variables.

How do you check if your solution to a word problem is correct?+

Substitute your answers back into the original conditions. Here, if silver = 4 points and gold = 6 points, then Bernie's total should be 4(4) + 2(6) = 28 ✓ and Kenny's should be 3(4) + 3(6) = 30 ✓. Both check out perfectly.

NJ

Neven Jurkovic, PhD

Professor of Computer Science, Palo Alto College, Alamo Colleges District, San Antonio, TX

Developer of Algebrator

Contact

This solution was prepared with AI assistance and reviewed by Dr. Jurkovic for mathematical accuracy and pedagogical clarity.

2026-07-05