Systems with Three Variables

Introduction

Many mathematical and real-world problems involve more than two unknown quantities.

For example, a school may sell three types of tickets, a business may manufacture three different products, or a scientist may combine three substances to produce a mixture.

In these situations, two variables may not be enough to describe all the unknown information.

A system of three-variable equations allows us to represent relationships involving three unknown quantities and determine values that satisfy all the equations simultaneously.

Three-variable systems extend the ideas already used in two-variable systems. We still define variables, construct equations, and look for values that make every equation true. However, the mathematics becomes more complex because we are working with three unknowns instead of two.

Three-variable systems are an important foundation for algebra, geometry, engineering, economics, chemistry, and more advanced mathematics.

What Is a Three-Variable System?

A three-variable system is a collection of equations involving three unknown quantities.

The variables are commonly represented by:

x, y, and z

For example:

x + y + z = 12

2x + y − z = 7

x − y + 2z = 7

These three equations form a system.

The goal is to find values of x, y, and z that satisfy all three equations at the same time.

A possible solution is written as an ordered triple:

(x, y, z)

For this system, the solution is:

(4, 5, 3)

We can verify this by substituting the values into each equation.

First equation:

4 + 5 + 3 = 12

Second equation:

2(4) + 5 − 3 = 10

This does not equal 7.

Therefore, (4, 5, 3) is not a solution to the system.

This illustrates an important principle: a solution must satisfy every equation, not just one or two of them.

We will return to the correct solution later.

Recognising Equations with Three Unknowns

An equation with three variables contains three different unknown quantities.

Examples include:

x + y + z = 20

2x − 3y + 4z = 15

5x + 2y − z = 8

3x + 6y + 2z = 30

Each equation contains x, y, and z.

However, an equation does not need to display all three variables explicitly to belong to a three-variable system.

For example:

x + y = 10

y + z = 12

x + z = 14

Together, these equations form a three-variable system because the system involves x, y, and z.

We can rewrite the first equation as:

x + y + 0z = 10

The coefficient of z is zero.

Identifying the variables

Consider:

4a + 2b − 3c = 18

The variables are:

• a

• b

• c

The coefficients are:

• 4 for a

• 2 for b

• −3 for c

The constant on the right side is 18.

Variable names do not have to be x, y, and z. They can represent quantities such as price, distance, time, or mass.

The General Form of a Linear Equation in Three Variables

A linear equation in three variables can be written as:

ax + by + cz = d

where:

• x, y, and z are variables.

• a, b, and c are coefficients.

• d is a constant.

• At least one of a, b, and c is nonzero.

For example:

2x + 3y − 4z = 12

Here:

a = 2

b = 3

c = −4

d = 12

The equation is linear because each variable appears only to the first power, and the variables are not multiplied together.

Linear and nonlinear equations

Three-variable systems can contain nonlinear equations, but this topic focuses on linear systems.

What Does a Solution Represent?

A solution to a three-variable system is an ordered triple of values that makes every equation true.

For example, consider:

x + y + z = 9

2x + y + z = 11

x + 2y + z = 12

We can test:

(x, y, z) = (2, 3, 4)

First equation:

2 + 3 + 4 = 9

Correct.

Second equation:

2(2) + 3 + 4 = 11

Correct.

Third equation:

2 + 2(3) + 4 = 12

Correct.

All three equations are satisfied.

Therefore:

(2, 3, 4) is a solution to the system.

The order matters

The ordered triple:

(2, 3, 4)

means:

x = 2

y = 3

z = 4

The triple (4, 3, 2) represents different values and is not necessarily a solution.

Always write the values in the order specified by the variables.

Why One Equation Is Usually Not Enough

Consider:

x + y + z = 12

Many different triples satisfy this equation.

Examples include:

(2, 4, 6)

(3, 5, 4)

(1, 1, 10)

(6, 3, 3)

Each triple adds to 12.

One equation generally does not provide enough information to determine three unknowns uniquely.

Adding a second equation

Suppose we also know:

x − y = 2

We now have:

x + y + z = 12

x − y = 2

There are still infinitely many real-number solutions.

For example:

(5, 3, 4)

(4, 2, 6)

(3, 1, 8)

Each triple satisfies both equations.

Adding a third independent equation

Suppose we also know:

y + z = 7

Our system becomes:

x + y + z = 12

x − y = 2

y + z = 7

The third equation provides additional information.

From:

x + y + z = 12

and:

y + z = 7

we obtain:

x = 5

Then:

5 − y = 2

y = 3

Finally:

3 + z = 7

z = 4

Therefore:

(x, y, z) = (5, 3, 4)

The system now has a unique solution.

Important qualification

Three equations do not automatically guarantee a unique solution.

Some equations may repeat information or contradict one another.

A three-variable system may have:

• Exactly one solution.

• No solution.

• Infinitely many solutions.

The number and independence of the equations determine the possible outcomes.

Comparing Two-Variable and Three-Variable Systems

Two-variable and three-variable systems follow the same general mathematical principles, but they differ in several important ways.

Two-variable example

x + y = 10

x − y = 2

Add the equations:

2x = 12

x = 6

Then:

y = 4

Solution:

(6, 4)

Three-variable example

x + y + z = 12

x − y = 2

y + z = 7

As shown earlier:

x = 5

y = 3

z = 4

Solution:

(5, 3, 4)

The central idea remains the same: find values that satisfy all equations simultaneously.

Graphical Meaning of Three-Variable Equations

A linear equation in two variables generally represents a straight line on a two-dimensional coordinate plane.

A linear equation in three variables represents a plane in three-dimensional space.

For example:

x + y + z = 6

represents a plane containing all points whose coordinates add to 6.

Some points on this plane include:

(6, 0, 0)

(0, 6, 0)

(0, 0, 6)

(2, 2, 2)

What is a plane?

A plane is a flat, two-dimensional surface extending infinitely in three-dimensional space.

A sheet of paper provides a useful physical analogy, although a mathematical plane has no thickness and extends without boundaries.

What happens when three planes intersect?

A system of three linear equations corresponds to three planes.

The solution consists of points that lie on all three planes simultaneously.

Depending on the arrangement, the system may have one solution, no solution, or infinitely many solutions.

A Unique Solution: Three Planes Meeting at One Point

Consider:

x = 2

y = 3

z = 4

Each equation represents a plane.

The plane x = 2 contains all points whose x-coordinate is 2.

The plane y = 3 contains all points whose y-coordinate is 3.

The plane z = 4 contains all points whose z-coordinate is 4.

All three planes share exactly one point:

(2, 3, 4)

This is a system with a unique solution.

Algebraic example

x + y + z = 9

2x + y + z = 11

x + 2y + z = 12

The unique solution is:

(2, 3, 4)

Graphically, the three planes share one common point.

No Solution: Planes with No Common Point

Consider:

x + y + z = 6

x + y + z = 10

The left sides are identical, but the right sides are different.

The same three values cannot add to both 6 and 10.

These equations represent parallel, distinct planes.

Adding a third equation cannot create a common point.

Therefore, the system has no solution.

Important idea

A system has no solution when there is no point that satisfies all its equations.

Not every inconsistent three-variable system consists of parallel planes. Three planes may also intersect pairwise without sharing a single common point.

Infinitely Many Solutions: Planes Sharing a Line

Consider:

x + y + z = 6

2x + 2y + 2z = 12

The second equation is twice the first.

Therefore, both equations represent the same plane.

Now add:

x − y = 0

This equation represents another plane.

The first and third planes intersect along a line.

Every point on that line satisfies all three equations.

For example:

x = y

and:

2x + z = 6

Let x = t.

Then:

y = t

z = 6 − 2t

The solutions have the form:

(t, t, 6 − 2t)

where t can be any real number.

Examples include:

(0, 0, 6)

(1, 1, 4)

(2, 2, 2)

(3, 3, 0)

There are infinitely many solutions.

Organising Information in Three-Variable Systems

One of the most important skills is translating written information into mathematical equations.

When three unknown quantities are involved, careful organisation becomes especially important.

Example: School Supplies

A school shop sells:

• Notebooks.

• Pens.

• Pencils.

A customer purchases:

2 notebooks, 3 pens, and 1 pencil for $15.

Another purchases:

1 notebook, 2 pens, and 4 pencils for $13.

A third purchases:

3 notebooks, 1 pen, and 2 pencils for $18.

We want to determine the price of each item.

Step 1: Define variables

n = price of one notebook in dollars

p = price of one pen in dollars

c = price of one pencil in dollars

Step 2: Organise the information

Step 3: Construct equations

First purchase:

2n + 3p + c = 15

Second purchase:

n + 2p + 4c = 13

Third purchase:

3n + p + 2c = 18

These equations form a three-variable system.

Step 4: Interpret the meaning

A solution (n, p, c) would give the individual prices of notebooks, pens, and pencils.

The system must satisfy all three purchases simultaneously.

We will solve this system in a later section.

Using Tables to Organise Information

Tables are particularly useful when word problems contain several categories.

For example, suppose a cinema sells adult, student, and child tickets.

Let:

a = number of adult tickets

s = number of student tickets

c = number of child tickets

If 200 tickets are sold:

a + s + c = 200

Suppose the prices are:

Adult: $15

Student: $10

Child: $6

If the total revenue is $2,200:

15a + 10s + 6c = 2200

Suppose 20 more adult tickets than child tickets were sold:

a − c = 20

We now have three equations:

a + s + c = 200

15a + 10s + 6c = 2200

a − c = 20

These equations represent three separate pieces of information.

The first describes the total number of tickets.

The second describes the total revenue.

The third describes the relationship between two ticket categories.

A useful habit is to identify what each equation represents before attempting to solve the system.

Worked Example: Finding Three Unknown Numbers

Three numbers have a sum of 24.

The first number is twice the second.

The third number is 3 greater than the second.

Find the three numbers.

Step 1: Define variables

x = first number

y = second number

z = third number

Step 2: Translate the statements

The sum is 24:

x + y + z = 24

The first is twice the second:

x = 2y

The third is 3 greater than the second:

z = y + 3

Step 3: Substitute

Substitute x = 2y and z = y + 3 into the first equation:

2y + y + (y + 3) = 24

4y + 3 = 24

4y = 21

y = 5.25

Step 4: Find the other values

x = 2(5.25) = 10.5

z = 5.25 + 3 = 8.25

Step 5: Verify

Sum:

10.5 + 5.25 + 8.25 = 24

First is twice the second:

10.5 = 2(5.25)

Third is 3 greater than the second:

8.25 = 5.25 + 3

Answer: The three numbers are 10.5, 5.25, and 8.25.

This example demonstrates that three-variable systems can be solved using substitution when two variables are easily expressed in terms of the third.

Worked Example: Three Types of Tickets

A school sells adult, student, and child tickets for a performance.

The prices are:

• Adult: $15

• Student: $10

• Child: $6

A total of 200 tickets are sold.

The total revenue is $2,200.

There are 20 more adult tickets than child tickets.

Find the number of each ticket type.

Step 1: Define variables

a = adult tickets

s = student tickets

c = child tickets

Step 2: Construct the system

Total tickets:

a + s + c = 200

Total revenue:

15a + 10s + 6c = 2200

Relationship:

a − c = 20

Step 3: Express a in terms of c

a = c + 20

Substitute into the total ticket equation:

(c + 20) + s + c = 200

s = 180 − 2c

Step 4: Substitute into the revenue equation

15(c + 20) + 10(180 − 2c) + 6c = 2200

Expand:

15c + 300 + 1800 − 20c + 6c = 2200

Combine:

c + 2100 = 2200

c = 100

Step 5: Find a and s

a = c + 20

a = 120

s = 180 − 2(100)

s = −20

Step 6: Interpret

The mathematical solution is:

a = 120

s = −20

c = 100

However, a negative number of student tickets is impossible.

Therefore, the conditions do not describe a valid real-world ticket sale.

Answer: There is no physically meaningful solution using nonnegative whole-number ticket quantities.

This is an important reminder that an algebraically consistent system may still produce values that violate contextual restrictions.

Worked Example: Solving a Three-Variable System by Elimination

Consider:

x + y + z = 12

2x − y + z = 7

x + 2y − z = 6

We want to determine x, y, and z.

Step 1: Label the equations

Equation A:

x + y + z = 12

Equation B:

2x − y + z = 7

Equation C:

x + 2y − z = 6

Step 2: Eliminate z from A and B

Subtract A from B:

(2x − y + z) − (x + y + z) = 7 − 12

x − 2y = −5

Call this Equation D.

Step 3: Eliminate z from A and C

Add A and C:

(x + y + z) + (x + 2y − z) = 12 + 6

2x + 3y = 18

Call this Equation E.

Step 4: Solve the two-variable system

We now have:

x − 2y = −5

2x + 3y = 18

Multiply the first equation by 2:

2x − 4y = −10

Subtract:

(2x + 3y) − (2x − 4y) = 18 − (−10)

7y = 28

y = 4

Step 5: Find x

x − 2(4) = −5

x − 8 = −5

x = 3

Step 6: Find z

Substitute into Equation A:

3 + 4 + z = 12

z = 5

Step 7: Verify

Equation A:

3 + 4 + 5 = 12

Equation B:

2(3) − 4 + 5 = 7

Equation C:

3 + 2(4) − 5 = 6

All three equations are satisfied.

Answer: (x, y, z) = (3, 4, 5).

Why elimination works

The original system has three unknowns.

By eliminating z from two pairs of equations, we create a smaller system containing only x and y.

After solving that system, we substitute the values back to find z.

This is the main strategy used for solving many three-variable linear systems.

Revisiting the Opening Example

At the beginning of these notes, we considered:

x + y + z = 12

2x + y − z = 7

x − y + 2z = 7

Let's determine the correct solution.

Step 1: Eliminate z

Add the first and second equations:

3x + 2y = 19

Multiply the first equation by 2:

2x + 2y + 2z = 24

Subtract the third equation:

x + 3y = 17

Step 2: Solve the resulting system

3x + 2y = 19

x + 3y = 17

From the second:

x = 17 − 3y

Substitute:

3(17 − 3y) + 2y = 19

51 − 9y + 2y = 19

−7y = −32

y = 32/7

Then:

x = 17 − 3(32/7)

x = 23/7

Step 3: Find z

x + y + z = 12

23/7 + 32/7 + z = 12

55/7 + z = 84/7

z = 29/7

Step 4: Verify

First equation:

23/7 + 32/7 + 29/7 = 84/7 = 12

Second equation:

2(23/7) + 32/7 − 29/7 = 49/7 = 7

Third equation:

23/7 − 32/7 + 2(29/7) = 49/7 = 7

Correct solution: (23/7, 32/7, 29/7).

Applications of Three-Variable Systems

Business and Economics

Businesses may need to determine the prices or quantities of three products using sales information.

Three-variable systems can also model production costs and financial relationships.

Chemistry

Scientists may use systems to determine the quantities of several substances in a mixture.

They can also use linear equations when balancing chemical reactions or analysing experimental data.

Engineering

Engineers use systems of equations to model forces, electrical circuits, structures, and mechanical systems.

Geometry

Three-variable equations represent planes in three-dimensional space.

Their intersections help describe geometric relationships.

Computer Science

Systems of equations are important in computer graphics, simulations, optimisation, and numerical modelling.

Transport and Logistics

Three-variable systems can describe quantities such as vehicle numbers, delivery costs, and transport capacities.

Common Mistakes

Assuming three variables always require exactly three equations

Three independent equations are typically sufficient for a unique solution, but three equations may be dependent or inconsistent. Some systems may require additional information.

Confusing an equation with a system

An equation is one mathematical relationship. A system contains multiple equations considered together.

Forgetting the order of the solution

The ordered triple (x, y, z) must match the variable definitions.

Checking only one equation

A valid solution must satisfy every equation.

Assuming all three-variable systems have one solution

A system may have one solution, no solution, or infinitely many solutions.

Incorrectly translating word problems

For example, "twice as many adults as children" means a = 2c, not a = c + 2.

Losing negative signs during elimination

When subtracting an equation, every term must be subtracted correctly.

Confusing three-dimensional coordinates with two-dimensional coordinates

A point in three-dimensional space has three coordinates, not two.

Accepting unrealistic contextual values

Negative numbers of tickets or fractional numbers of people may not make sense.

Failing to organise information

Tables, variable definitions, and labelled equations help prevent errors.

Key Terms

• System of equations: Two or more equations considered simultaneously.

• Three-variable system: A system involving three unknown quantities.

• Variable: A symbol representing an unknown quantity.

• Coefficient: A number multiplying a variable.

• Constant: A fixed numerical value.

• Linear equation: An equation in which variables occur only to the first power and are not multiplied together.

• Solution: Values that satisfy all equations in a system.

• Ordered triple: Three values written in a specified order, such as (x, y, z).

• Independent equations: Equations that provide distinct mathematical information.

• Dependent equations: Equations containing redundant information.

• Consistent system: A system with at least one solution.

• Inconsistent system: A system with no solution.

• Unique solution: Exactly one solution.

• Infinitely many solutions: A system with an unlimited number of solutions.

• Plane: A flat, two-dimensional surface in three-dimensional space.

• Intersection: A point or set of points shared by geometric objects.

• Substitution: Replacing a variable with an equivalent expression.

• Elimination: Combining equations to remove a variable.

• Verification: Checking that proposed values satisfy the original equations.

• Constraint: A restriction on the possible values of variables.

Key Takeaways

• Three-variable systems involve three unknown quantities, commonly x, y, and z.

• A linear equation in three variables has the general form ax + by + cz = d.

• A solution is an ordered triple that satisfies every equation.

• One or two equations generally do not determine three unknowns uniquely.

• Three independent, consistent linear equations can determine a unique solution.

• A system may have one solution, no solution, or infinitely many solutions.

• Linear equations in three variables represent planes in three-dimensional space.

• Tables and clearly defined variables help organise information from word problems.

• Elimination can reduce a three-variable system to a two-variable system.

• Every solution should be verified by substitution into the original equations.

• Contextual restrictions must be considered when interpreting real-world answers.

Check Your Understanding

Recall and Understanding

1. What is a three-variable system?

2. Write the general form of a linear equation in three variables.

3. What does the ordered triple (2, 5, 7) represent?

4. Explain why one linear equation in three variables generally does not determine a unique solution.

5. Describe the graphical difference between two-variable and three-variable linear systems.

Identifying and Organising Systems

6. Identify the variables and coefficients in:

3x − 2y + 5z = 18

7. Determine whether the following equations are linear:

a. x + 2y − z = 9

b. x² + y + z = 10

c. 3x + 4y = 12

d. xy + z = 7

8. A shop sells three types of stationery. Two notebooks, one pen, and three pencils cost $14. One notebook, three pens, and two pencils cost $13. Three notebooks, two pens, and one pencil cost $19. Define variables and construct the system.

9. Explain the difference between a consistent and an inconsistent system.

10. Can three linear equations in three variables have infinitely many solutions? Explain.

Application

11. Determine whether (2, 3, 4) satisfies:

x + y + z = 9

2x + y − z = 3

x − y + 2z = 7

12. Solve:

x + y + z = 15

x − y = 3

y + z = 9

13. Solve:

x + y + z = 10

2x + y + z = 13

x + 2y + z = 14

14. Determine whether the system has a solution:

x + y + z = 8

2x + 2y + 2z = 20

x − y = 3

15. A school sells adult, student, and child tickets. A total of 100 tickets are sold. Adult tickets cost $12, student tickets cost $8, and child tickets cost $5. The total revenue is $850. There are 10 more adult tickets than child tickets. Construct the system and determine the number of each ticket type.

Answers to Check Your Understanding

1. A three-variable system is a collection of equations involving three unknown quantities that must be satisfied simultaneously.

2.

ax + by + cz = d

At least one of a, b, and c must be nonzero.

3.

x = 2

y = 5

z = 7

The order of the values matters.

4. One equation usually allows infinitely many combinations of three values. Additional independent relationships are needed to determine a unique solution.

5. A linear equation in two variables represents a line in two-dimensional space. A linear equation in three variables represents a plane in three-dimensional space.

6.

Variables:

x, y, z

Coefficients:

3, −2, 5

Constant:

18

7.

a. Linear

b. Nonlinear, because x is squared.

c. Linear; the coefficient of z is zero.

d. Nonlinear, because x and y are multiplied.

8.

Let:

n = price of one notebook

p = price of one pen

c = price of one pencil

The system is:

2n + p + 3c = 14

n + 3p + 2c = 13

3n + 2p + c = 19

9. A consistent system has at least one solution. An inconsistent system has no solution.

10. Yes. If the equations are dependent or share a common line or plane, infinitely many solutions may exist.

11.

Test (2, 3, 4).

First equation:

2 + 3 + 4 = 9

Correct.

Second equation:

2(2) + 3 − 4 = 3

Correct.

Third equation:

2 − 3 + 2(4) = 7

Correct.

Answer: Yes, (2, 3, 4) satisfies the system.

12.

x + y + z = 15

x − y = 3

y + z = 9

Subtract the third equation from the first:

x = 6

Then:

6 − y = 3

y = 3

Finally:

3 + z = 9

z = 6

Answer: (6, 3, 6).

Verification:

6 + 3 + 6 = 15

6 − 3 = 3

3 + 6 = 9

13.

x + y + z = 10

2x + y + z = 13

x + 2y + z = 14

Subtract the first equation from the second:

x = 3

Subtract the first equation from the third:

y = 4

Substitute:

3 + 4 + z = 10

z = 3

Answer: (3, 4, 3).

14.

First equation:

x + y + z = 8

Multiply by 2:

2x + 2y + 2z = 16

However, the second equation states:

2x + 2y + 2z = 20

This is a contradiction.

Answer: The system has no solution.

15.

Let:

a = adult tickets

s = student tickets

c = child tickets

The equations are:

a + s + c = 100

12a + 8s + 5c = 850

a − c = 10

From the third equation:

a = c + 10

Substitute into the first:

c + 10 + s + c = 100

s = 90 − 2c

Substitute into the revenue equation:

12(c + 10) + 8(90 − 2c) + 5c = 850

12c + 120 + 720 − 16c + 5c = 850

c + 840 = 850

c = 10

Then:

a = 20

s = 90 − 20 = 70

Verification:

20 + 70 + 10 = 100

12(20) + 8(70) + 5(10)

= 240 + 560 + 50

= $850

Adult tickets exceed child tickets by:

20 − 10 = 10

Answer: 20 adult tickets, 70 student tickets, and 10 child tickets.

Final Summary

Three-variable systems extend the principles of simultaneous equations to situations involving three unknown quantities.

These systems can be represented algebraically using equations such as ax + by + cz = d, and graphically using planes in three-dimensional space.

A solution is an ordered triple that satisfies every equation. Depending on the relationships between the equations, a system may have one solution, no solution, or infinitely many solutions.

The central idea is that each equation provides information about the unknown quantities, and a valid solution must satisfy all the information simultaneously.