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Setting Up
Linear Systems

Learn how to translate equations and word problems into systems of linear equations: the essential first step to solving them.

Linear Algebra

What Is a Linear System?

A system of linear equations is a set of two or more linear equations that share the same variables. The goal is to find values for those variables that satisfy every equation simultaneously.

Key Vocabulary

Variable
An unknown quantity you solve for, such as \( x \), \( y \), or \( z \).
Coefficient
The number multiplied by a variable: in \( 3x \), the coefficient is 3.
Constant
A fixed number with no variable attached, like the 10 in \( x + y = 10 \).

Why Setting Up Matters

Before you can solve a system, you must set it up correctly. Most errors happen at this stage: wrong variables, missing constraints, or misread word problems. A perfect solution method applied to a wrong setup still gives a wrong answer.

Real-World Use
Budget planning, mixing solutions, speed and distance, supply and demand: all modeled by linear systems.
Foundation Skill
Substitution, elimination, and matrix methods all require a correctly written system as input.

Standard Form of a Linear System

\( a_1 x + b_1 y = c_1 \)
\( a_2 x + b_2 y = c_2 \)
Two equations, two unknowns (2 × 2 system)

Graphical Interpretation

x y
\( x + y = 10 \)
\( x - y = 4 \)
\( (7, 3) \)

Equations vs. Word Problems

From Equations From Word Problems
Starting Point Equations already given A story or scenario in English
Main Task Rewrite in standard form Define variables, then write equations
Common Pitfall Forgetting to align terms Mis-identifying what the variables represent
Difficulty Lower: mechanical rearranging Higher: requires reading comprehension

Step-by-Step Setup Process

1
Define your variables. State clearly what each letter represents, including units.
2
Identify the relationships. Find two (or more) distinct conditions connecting the variables.
3
Write each relationship as an equation. Translate each condition into a linear equation.
4
Align in standard form. Rearrange so all variable terms are on the left and constants on the right.
5
Check: do you have enough equations? You need at least as many equations as unknowns.

Example 1: From Equations

Rewrite the following pair as a system in standard form:

1
Rearrange equation 1: \( y = 2x + 3 \;\Rightarrow\; -2x + y = 3 \)
2
Rearrange equation 2: \( y = -x + 9 \;\Rightarrow\; x + y = 9 \)
3
System: \( -2x + y = 3 \) and \( x + y = 9 \)

Example 2: Word Problem

1
Define variables: Let \( p \) = number of pens, \( n \) = number of notebooks.
2
Total items: \( p + n = 8 \)
3
Total cost: \( 2p + 5n = 25 \)

Example 3: Speed and Distance

1
Define: Let \( b \) = boat speed (km/h), \( c \) = current speed (km/h).
2
Downstream: effective speed = \( b + c \). Distance = rate \(\times\) time: \( 2(b + c) = 60 \;\Rightarrow\; b + c = 30 \)
3
Upstream: effective speed = \( b - c \). \( 4(b - c) = 40 \;\Rightarrow\; b - c = 10 \)

Extending to Three Variables

Systems can have more than two unknowns. A 3 \(\times\) 3 system requires three independent equations to find a unique solution.

\( 2x + y - z = 3 \)
\( x - 3y + 2z = 1 \)
\( 3x + 2y + z = 10 \)
Three equations, three unknowns (3 × 3 system)

The Matrix Connection

Once your system is in standard form, it can be expressed as a matrix equation \( A\mathbf{x} = \mathbf{b} \). This is the gateway to powerful solving methods like Gaussian elimination.

\( \begin{bmatrix} 2 & 1 & -1 \\ 1 & -3 & 2 \\ 3 & 2 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 3 \\ 1 \\ 10 \end{bmatrix} \)

Word Problem Translation Tips

English Phrase Math Translation
"sum of" / "total" \( + \) (addition)
"difference" / "less than" \( - \) (subtraction)
"times" / "each" / "per" \( \times \) (multiplication)
"is" / "equals" / "was" \( = \)
"twice" / "double" \( 2 \cdot \) (variable)
"more than" \( + \) (added to variable)

Common Mistakes

✕ Undefined Variables
Jumping straight to equations without stating what each variable represents leads to confusion and wrong setups.
✕ Redundant Equations
Writing the same relationship twice in different forms: e.g. \( x + y = 10 \) and \( 2x + 2y = 20 \). You need independent equations.
✕ Swapped Coefficients
Mixing up which quantity gets which coefficient: "pens at $2 each" means \( 2p \), not \( p \cdot \text{(price of notebooks)} \).

Setup Checklist Summary

Step From Equations From Word Problems
1. Variables Already given Define and label clearly
2. Relationships Equations are the relationships Extract from story sentences
3. Standard Form Move all terms: \( ax + by = c \) Translate phrases into \( ax + by = c \)
4. Count Check Equations >= unknowns? Equations >= unknowns?
5. Verify Plug in a test value Re-read problem, check logic

Question 1 of 5

Which of the following is a system of linear equations?

Question 2 of 5

"The sum of two numbers is 20. Their difference is 6." Which system represents this?

Question 3 of 5

Rewrite \( y = 3x - 1 \) in standard form \( ax + by = c \). Which is correct?

Question 4 of 5

"Adult tickets cost $8 and child tickets cost $5. A family buys 6 tickets for $39." Which system models this?

Question 5 of 5

A system has 3 unknowns. What is the minimum number of independent equations needed for a unique solution?