🧊 Algebra Solver

System of 3 Equations Solver

Solve systems of three linear equations with three variables step-by-step using elimination, matrices, or Cramer’s rule.

System of 3 Equations

Enter values to calculate instant result

Equation 1 ℹ️ Info

Equation 2 ℹ️ Info

Equation 3 ℹ️ Info

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How to Use & Educational Guide

In-depth guide for System of 3 Equations Solver

Educational Overview

What is a System of 3 Equations?

A system of 3 equations is a set of three equations that contain three unknown variables, usually $x$, $y$, and $z$. Solving the system means finding the values of $x$, $y$, and $z$ that make all three equations true at the same time. For a linear system, the equations can be written in the form $a_1x+b_1y+c_1z=d_1$, $a_2x+b_2y+c_2z=d_2$, and $a_3x+b_3y+c_3z=d_3$.

System of 3 Equations Formula

A standard system of three linear equations with three variables can be represented as:

$$a_1x+b_1y+c_1z=d_1$$

$$a_2x+b_2y+c_2z=d_2$$

$$a_3x+b_3y+c_3z=d_3$$

These equations can also be written as an augmented matrix:

$$\left[\begin{array}{ccc|c} a_1 & b_1 & c_1 & d_1 \ a_2 & b_2 & c_2 & d_2 \ a_3 & b_3 & c_3 & d_3 \end{array}\right]$$

A system may have one unique solution, no solution, or infinitely many solutions depending on the relationships between the equations.

How to Solve a System of 3 Equations Step-by-Step?

The elimination method is one of the most useful ways to solve a system of three linear equations. The goal is to reduce the original system to a simpler 2-variable system.

Step 1 (Write the Equations in Standard Form): Arrange each equation as $ax+by+cz=d$ and identify the coefficients of $x$, $y$, and $z$.
Step 2 (Eliminate One Variable): Combine pairs of equations to eliminate one variable, such as $z$.
Step 3 (Create a 2×2 System): The elimination produces two equations containing only $x$ and $y$.
Step 4 (Solve for x and y): Solve the resulting 2-variable system using elimination or substitution.
Step 5 (Back-Substitute for z): Substitute the known values of $x$ and $y$ into one original equation to find $z$.
Step 6 (Verify the Solution): Substitute $x$, $y$, and $z$ into all three original equations to confirm that every equation is satisfied.

Methods for Solving 3 Equations

Elimination Method

Eliminate one variable from two pairs of equations, solve the resulting 2-variable system, and then back-substitute.

Matrix Method

Represent the equations as an augmented matrix and use Gaussian elimination to reduce the matrix and find the unknown variables.

Cramer’s Rule

Use determinants to calculate $x$, $y$, and $z$. This method works directly when the coefficient determinant is nonzero.

System of 3 Equations Examples

Example 1: Solve a System of 3 Equations

Eq 1: $x+y+z=6$
Eq 2: $2x-y+z=3$
Eq 3: $x+2y-z=2$

• Step 1: Add Eq 1 and Eq 3 to eliminate $z$: $2x+3y=8$.

• Step 2: Add Eq 2 and Eq 3 to eliminate $z$: $3x+y=5$.

• Step 3: From $3x+y=5$, $y=5-3x$. Substitute into $2x+3y=8$:

$2x+3(5-3x)=8 \implies -7x=-7 \implies \mathbf{x=1}$.

• Step 4: $y=5-3(1)=\mathbf{2}$.

• Step 5: Substitute into Eq 1: $1+2+z=6 \implies \mathbf{z=3}$.

• Check: All three original equations are satisfied.

👉 Solution: $(x,y,z)=(1,2,3)$

Example 2: System with a Missing Variable

Eq 1: $x+y=3$
Eq 2: $2x+z=5$
Eq 3: $y+z=4$

• Step 1: From Eq 1, $y=3-x$.

• Step 2: From Eq 3, $z=4-y=4-(3-x)=1+x$.

• Step 3: Substitute into Eq 2: $2x+(1+x)=5$.

$3x+1=5 \implies 3x=4 \implies \mathbf{x=\frac{4}{3}}$.

• Step 4: $y=3-\frac{4}{3}=\mathbf{\frac{5}{3}}$.

• Step 5: $z=4-\frac{5}{3}=\mathbf{\frac{7}{3}}$.

👉 Solution: $\left(x,y,z\right)=\left(\frac{4}{3},\frac{5}{3},\frac{7}{3}\right)$

Example 3: System with a Negative Coefficient

Eq 1: $x+y+z=4$
Eq 2: $2x-y+z=1$
Eq 3: $x+2y-z=4$

• Step 1: Add Eq 1 and Eq 3: $2x+3y=8$.

• Step 2: Add Eq 2 and Eq 3: $3x+y=5$.

• Step 3: Solving the reduced equations gives $x=\frac{7}{7}=1$ and $y=2$.

• Step 4: Substitute into Eq 1: $1+2+z=4 \implies \mathbf{z=1}$.

👉 Solution: $(x,y,z)=(1,2,1)$

Real-World Applications of 3-Equation Systems

⚡ Electrical Circuits

Engineers can use simultaneous equations from Kirchhoff’s circuit laws to determine unknown currents and voltages in electrical networks.

📊 Business & Economics

Systems of equations can model relationships between several unknown quantities, such as prices, quantities, costs, or production levels.

Common Mistakes to Avoid

Ignoring Missing Variables: If an equation has no $y$ term, its coefficient is $0$, not $1$. For example, $2x+z=5$ is $2x+0y+z=5$.
Sign Errors: Be especially careful when subtracting equations containing negative coefficients.
Incorrect Elimination: When multiplying an equation before elimination, multiply every term on both sides by the same number.
Skipping Verification: Always substitute the final values of $x$, $y$, and $z$ into all three original equations.

Mathematical accuracy verified by AptCalc Engine

Frequently Asked Questions (FAQs)

Answers to common questions about this calculator

How do you solve a system of 3 equations?

A system of 3 linear equations can be solved by eliminating one variable to create a 2-variable system, solving for two variables, and then back-substituting to find the third variable.

What is a system of 3 equations with 3 variables?

It is a set of three equations containing three unknowns, usually x, y, and z. A solution is a set of values for x, y, and z that satisfies all three equations.

Can I solve 3 equations using a matrix?

Yes. A 3-equation system can be written as an augmented matrix and solved using Gaussian elimination or other matrix methods.

Can a system of 3 equations have no solution?

Yes. If the equations are inconsistent, the system has no solution. A system can also have infinitely many solutions when the equations are dependent.