Systems Decoded Review Handout Finite Mathematics • Pre-Quiz Review
Systems of Linear Equations Reference
Lessons 3-4, 3-5, 3-6
Quick Study Guide
A system of linear equations is a collection of two or more equations with the same set of variables. The solution is the point or values that make all equations true simultaneously.
1
Solving by Graphing (Slope-Intercept Form: \(y = mx + b\))
Intersection \((x, y)\)
1. Find Slope & Intercept: For each equation \(y = mx + b\), identify the \(y\)-intercept \((0, b)\) and the slope \(m = \frac{\text{rise}}{\text{run}}\).
2. Plot Both Lines: Plot \((0, b)\) on the coordinate plane. Use the rise and run to plot at least 2 more points, then draw a straight line through them with arrows.
3. State the Intersection: Read the exact coordinate point \((x, y)\) where the two lines cross.
Quick Graphing Checklist
• Negative slope falls from left to right
• Positive slope rises from left to right
• Horizontal axis is \(x\); vertical axis is \(y\)
• Check by substituting \((x, y)\) into both lines!
2
Solving by Elimination (Single-Multiplier Strategy)
\(Ax + By = C\)
Step 1 (Target Opposite Terms): Look for coefficients that are easily made opposites by multiplying only one of the equations by a constant.
Step 2 (Multiply One Equation): Multiply every term in that equation (including the constant on the right side) by the factor.
Step 3 (Add & Solve): Add the two equations vertically to eliminate one variable. Solve for the remaining variable.
Step 4 (Back-Substitute): Plug your value into either original equation to find the second variable. Write your final answer as an ordered pair \((x, y)\).
Worked Mini-Model
\(\begin{cases} 4x + 3y = 24 \\ 2x - y = 2 \end{cases}\)
Multiply Eq 2 by 3 to eliminate \(y\):
\(\begin{aligned} 4x + 3y &= 24 \\ 6x - 3y &= 6 \end{aligned}\)
Add: \(10x = 30 \implies \mathbf{x = 3}\)
\(2(3) - y = 2 \implies \mathbf{y = 4}\)
Solution: \((3, 4)\)
Pro Tip: When plotting lines, always use 3 points to confirm they form a perfect straight line!
Page 1 of 2
Desmos Matrix Calculator & Applications
Methods 3, 4, 5 • desmos.com/matrix
3
Entering & Solving with Desmos Matrix
rref Function
1. Set Up Matrix [A|B]
For a 3-variable system, create a 3 × 4 matrix in Desmos. Make sure all equations are in standard form (\(x, y, z = \text{constant}\)).
Alert: If a variable is missing, enter 0 in that column!
2. Run rref Command
Click rref on the calculator keypad, type the matrix name (e.g., rref(A)), and hit Enter.
Reduced Row Echelon Form
3. Read the Variables
Read straight across each row:
Row 1: \([1\;0\;0 \mid k_1] \implies x = k_1\)
Row 2: \([0\;1\;0 \mid k_2] \implies y = k_2\)
Row 3: \([0\;0\;1 \mid k_3] \implies z = k_3\)
4
Word Problems: Finding the Price of 3 Items
Applied Matrix Modeling
Step 1: Variables
Define the price of each item:
Step 2: 3 Equations
Each order sentence forms one equation:
Step 3: Augmented Matrix
Strip the variables and place coefficients in a \(3 \times 4\) matrix. Run rref.
Step 4: Context Answer
Write your final answer in a full sentence with dollar signs ($) and units.
Example Matrix Translation:
\(\begin{cases} 3m + 4b + 2c = 23.00 \\ 2m + 5b + 3c = 27.00 \\ 4m + 2b + 5c = 30.50 \end{cases}\)
\(\left[\begin{array}{ccc|c} 3 & 4 & 2 & 23.00 \\ 2 & 5 & 3 & 27.00 \\ 4 & 2 & 5 & 30.50 \end{array}\right] \xrightarrow{\text{rref}} \begin{aligned} m &= \$2.00 \\ b &= \$2.50 \\ c &= \$3.50 \end{aligned}\)
High-Yield Quiz Reminders
• Reading \(y\) from rref: Look at Row 2, Column 4! Row 2 gives \(0x + 1y + 0z = k \implies y = k\).
• Missing Variables: If an equation says \(3x + 4z = 18\), remember the \(y\)-coefficient is 0 in the matrix!
Finite Mathematics • Systems of Linear Equations Page 2 of 2
Systems Mastery Quiz Finite Mathematics • Assessment Version A
Quiz: Systems of Linear Equations
Lessons 3-4, 3-5, 3-6 • Graphical, Algebraic, & Matrix Methods
Score: ________
Name:
Date:
Period:
1
Solve by Graphing
Both lines in slope-intercept form
Solve the system of linear equations by graphing both lines on the coordinate grid below. State your solution as an ordered pair \((x, y)\).
\(\text{Line 1: } y = \frac{1}{2}x + 1\)
\(\text{Line 2: } y = -x + 4\)
Solution \((x, y)\):
x y -6 -4 -2 2 4 6 6 4 2 -2 -4 -6 0 Grid: -8 to 8 on both axes • 1 grid square = 1 unit
2
Solve by Elimination
Show all algebraic steps
Solve the system of linear equations using the elimination method. Show all work and state your solution:
\(\begin{cases} 4x + 3y = 24 \\ 2x - y = 2 \end{cases}\)
Work Space:
Solution \((x, y)\):
Finite Mathematics • Quiz (Version A) Page 1 of 3
Student Name:
Augmented Matrices & Desmos Calculator (Version A)
3
Interpreting Reduced Row Echelon Form
3-Variable System
A system of linear equations in three variables \((x, y, z)\) was entered into the Desmos Matrix calculator. The resulting rref output is shown below:
\(\operatorname{rref}(A) = \left[\begin{array}{ccc|c} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & 5 \\ 0 & 0 & 1 & 3 \end{array}\right]\)
Based on this calculator output, what is the solution for \(y\)?
4
Solving a System with a Missing Variable
Use Desmos Matrix Calculator
Given the following system of linear equations:
\(\begin{cases} 2x + 3y - z = 10 \\ 3x + 4z = 18 \\ -x + 2y + z = 7 \end{cases}\)
(a) Write the augmented matrix for this system:
(b) State the solution:
Solve using Desmos and state variable values:
\(x = \underline{\hspace{55px}}\)
\(y = \underline{\hspace{55px}}\)
\(z = \underline{\hspace{55px}}\)
Finite Mathematics • Quiz (Version A) Page 2 of 3
Student Name:
Applied Finite Mathematics Modeling (Version A)
5
Matrix Modeling: Select the 3 Correct Rows
Multiple Choice by Row
The Apex Theater sold 400 total tickets for a play, generating $4,550 in total revenue. Adult tickets cost
Systems Mastery Answer Key Teacher Edition • Complete Solution Key Version A
Systems Mastery Quiz: Answer Key
Finite Mathematics • Lessons 3-4, 3-5, 3-6
Teacher Copy
Full Worked Solutions
1
Problem 1: Solving by Graphing
Solution
Line Analysis:
Line 1: \(y = \frac{1}{2}x + 1 \implies m = \frac{1}{2}, b = 1\)
Line 2: \(y = -x + 4 \implies m = -1, b = 4\)
Verification Check:
• Line 1: \(2 = \frac{1}{2}(2) + 1 \implies 2 = 2\) ✓
• Line 2: \(2 = -(2) + 4 \implies 2 = 2\) ✓
Solution:
\((x, y) = (2, 2)\)
(2, 2) Line 1: y = ½x + 1 Line 2: y = -x + 4
• Blue: Line 1 • Green: Line 2 • Red: (2, 2)
2
Problem 2: Elimination Method
\(\begin{cases} 4x + 3y = 24 \\ 2x - y = 2 \end{cases}\)
Solution Steps (Multiply Equation 2 by 3):
• Eq 1: \(4x + 3y = 24\)
• 3 × Eq 2: \(6x - 3y = 6\)
Add equations: \(10x = 30 \implies \mathbf{x = 3}\)
Back-substitute \(x = 3\) into Eq 2: \(2(3) - y = 2 \implies 6 - y = 2 \implies \mathbf{y = 4}\)
Alternative Method: Multiply Eq 2 by \(-2\) to eliminate \(x\): \(-4x + 2y = -4\). Adding yields \(5y = 20 \implies y = 4\).
Solution:
\((x, y) = (3, 4)\)
Systems Mastery Quiz • Answer Key (Version A) Page 1 of 4
Answer Key • Problems 3 & 4 (Version A)
Teacher Solutions
3
Problem 3: Interpreting rref Output
Reading Variable y
\(\operatorname{rref}(A) = \left[\begin{array}{ccc|c} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & 5 \\ 0 & 0 & 1 & 3 \end{array}\right]\)
Solution:
\(y = 5\)
Explanation: Row 2 translates to \(0x + 1y + 0z = 5 \implies y = 5\).
4
Problem 4: Missing Variable Matrix & Solution
Zero in y-column
(a) Augmented Matrix (Note 0 in Row 2, Col 2):
\(\left[\begin{array}{ccc|c} 2 & 3 & -1 & 10 \\ 3 & \mathbf{0} & 4 & 18 \\ -1 & 2 & 1 & 7 \end{array}\right]\)
(b) Solution:
\(\mathbf{x = 2}, \quad \mathbf{y = 3}, \quad \mathbf{z = 3}\)
Ordered Triple: \((2, 3, 3)\)
Systems Mastery Quiz • Answer Key (Version A) Page 2 of 4
Answer Key • Problems 5 & 6 (Version A)
Teacher Solutions
5
Problem 5: Matrix Row Selection Key
3 Correct Rows
Row 1 (Total Ticket Count): Equation: \(x + y + z = 400\)
Systems Mastery Quiz Form B Finite Mathematics • Assessment Version B
Quiz: Systems of Linear Equations
Lessons 3-4, 3-5, 3-6 • Graphical, Algebraic, & Matrix Methods
Score: ________
Name:
Date:
Period:
1
Solve by Graphing
Both lines in slope-intercept form
Solve the system of linear equations by graphing both lines on the coordinate grid below. State your solution as an ordered pair \((x, y)\).
\(\text{Line 1: } y = -\frac{1}{2}x + 2\)
\(\text{Line 2: } y = x - 4\)
Solution \((x, y)\):
x y -6 -4 -2 2 4 6 6 4 2 -2 -4 -6 0 Grid: -8 to 8 on both axes • 1 grid square = 1 unit
2
Solve by Elimination
Show all algebraic steps
Solve the system of linear equations using the elimination method. Show all work and state your solution:
\(\begin{cases} 3x + 2y = 18 \\ x - 4y = -8 \end{cases}\)
Work Space:
Solution \((x, y)\):
Finite Mathematics • Quiz (Version B) Page 1 of 3
Student Name:
Augmented Matrices & Desmos Calculator (Version B)
3
Interpreting Reduced Row Echelon Form
3-Variable System
A system of linear equations in three variables \((x, y, z)\) was entered into the Desmos Matrix calculator. The resulting rref output is shown below:
\(\operatorname{rref}(A) = \left[\begin{array}{ccc|c} 1 & 0 & 0 & 7 \\ 0 & 1 & 0 & -3 \\ 0 & 0 & 1 & 4 \end{array}\right]\)
Based on this calculator output, what is the solution for \(y\)?
4
Solving a System with a Missing Variable
Use Desmos Matrix Calculator
Given the following system of linear equations:
\(\begin{cases} 2x + y - 3z = 5 \\ 4x - z = 11 \\ -x + 3y + 2z = 5 \end{cases}\)
(a) Write the augmented matrix for this system:
(b) State the solution:
Solve using Desmos and state variable values:
\(x = \underline{\hspace{55px}}\)
\(y = \underline{\hspace{55px}}\)
\(z = \underline{\hspace{55px}}\)
Finite Mathematics • Quiz (Version B) Page 2 of 3
Student Name:
Applied Finite Mathematics Modeling (Version B)
5
Matrix Modeling: Select the 3 Correct Rows
Multiple Choice by Row
The Grand Theater sold 500 total tickets for a musical, generating $6,060 in total revenue. Adult tickets cost
Systems Mastery Quiz Form C Finite Mathematics • Assessment Version C
Quiz: Systems of Linear Equations
Lessons 3-4, 3-5, 3-6 • Graphical, Algebraic, & Matrix Methods
Score: ________
Name:
Date:
Period:
1
Solve by Graphing
Both lines in slope-intercept form
Solve the system of linear equations by graphing both lines on the coordinate grid below. State your solution as an ordered pair \((x, y)\).
\(\text{Line 1: } y = -\frac{2}{3}x + 3\)
\(\text{Line 2: } y = x - 2\)
Solution \((x, y)\):
x y -6 -4 -2 2 4 6 6 4 2 -2 -4 -6 0 Grid: -8 to 8 on both axes • 1 grid square = 1 unit
2
Solve by Elimination
Show all algebraic steps
Solve the system of linear equations using the elimination method. Show all work and state your solution:
\(\begin{cases} 2x + 5y = 19 \\ 6x - y = 9 \end{cases}\)
Work Space:
Solution \((x, y)\):
Finite Mathematics • Quiz (Version C) Page 1 of 3
Student Name:
Augmented Matrices & Desmos Calculator (Version C)
3
Interpreting Reduced Row Echelon Form
3-Variable System
A system of linear equations in three variables \((x, y, z)\) was entered into the Desmos Matrix calculator. The resulting rref output is shown below:
\(\operatorname{rref}(A) = \left[\begin{array}{ccc|c} 1 & 0 & 0 & -4 \\ 0 & 1 & 0 & 8 \\ 0 & 0 & 1 & -1 \end{array}\right]\)
Based on this calculator output, what is the solution for \(y\)?
4
Solving a System with a Missing Variable
Use Desmos Matrix Calculator
Given the following system of linear equations:
\(\begin{cases} 3x + y - 2z = 3 \\ 5x + 3z = 11 \\ -2x + 2y + z = 8 \end{cases}\)
(a) Write the augmented matrix for this system:
(b) State the solution:
Solve using Desmos and state variable values:
\(x = \underline{\hspace{55px}}\)
\(y = \underline{\hspace{55px}}\)
\(z = \underline{\hspace{55px}}\)
Finite Mathematics • Quiz (Version C) Page 2 of 3
Student Name:
Applied Finite Mathematics Modeling (Version C)
5
Matrix Modeling: Select the 3 Correct Rows
Multiple Choice by Row
The City Playhouse sold 350 total tickets for a musical, generating $3,270 in total revenue. Adult tickets cost
Systems Practice Review Worksheet Finite Mathematics • Pre-Quiz Practice Practice Worksheet
Systems of Equations Review
Lessons 3-4, 3-5, 3-6 • Graphical, Algebraic, & Matrix Practice
Quiz Prep
Show all steps
Name:
Date:
Period:
1
Solve by Graphing
Both lines in slope-intercept form
Solve the system of linear equations by graphing both lines on the coordinate grid below. State your solution as an ordered pair \((x, y)\).
\(\text{Line 1: } y = \frac{1}{3}x + 2\)
\(\text{Line 2: } y = -x + 6\)
Solution \((x, y)\):
x y -6 -4 -2 2 4 6 6 4 2 -2 -4 -6 0 Grid: -8 to 8 on both axes • 1 grid square = 1 unit
2
Solve by Elimination
Show all algebraic steps
Solve the system of linear equations using the elimination method. Show all work and state your solution:
\(\begin{cases} 2x + 3y = 18 \\ x - 2y = -5 \end{cases}\)
Work Space:
Solution \((x, y)\):
Finite Mathematics • Practice Review Worksheet Page 1 of 2
Student Name:
Augmented Matrices & Modeling Practice
3
Solving a System with a Missing Variable
Use Desmos Matrix Calculator
Given the following system of linear equations:
\(\begin{cases} 3x + 2y - z = 5 \\ 2x + 3z = 13 \\ -x + 4y + z = 5 \end{cases}\)
(a) Write the augmented matrix for this system:
(b) State the solution:
Solve using Desmos and state variable values:
\(x = \underline{\hspace{55px}}\)
\(y = \underline{\hspace{55px}}\)
\(z = \underline{\hspace{55px}}\)
4
Matrix Modeling Word Problem
Write Matrix & Solve
Community Theater Ticket Sales: A local theater sold 300 total tickets for their opening night performance, generating $3,100 in total revenue. Adult tickets cost $14 , student tickets cost $8 , and child tickets cost $6 . The theater sold 20 more student tickets than child tickets .
(a) Write the augmented matrix for this system (in \(x, y, z\) order):
(b) State the number of tickets sold:
Solve using Desmos Matrix calculator:
Adult (\(x\)): ______ tickets
Systems Practice Review Key Teacher Edition • Practice Review Key
Systems Practice Review: Solutions
Finite Mathematics • Lessons 3-4, 3-5, 3-6
Worked Key
Practice Problems 1–4
1
Problem 1: Solving by Graphing
Solution
Line Analysis:
Line 1: \(y = \frac{1}{3}x + 2 \implies m = \frac{1}{3}, b = 2\)
Line 2: \(y = -x + 6 \implies m = -1, b = 6\)
Verification Check:
• Line 1: \(3 = \frac{1}{3}(3) + 2 \implies 3 = 3\) ✓
• Line 2: \(3 = -(3) + 6 \implies 3 = 3\) ✓
Solution:
\((x, y) = (3, 3)\)
(3, 3) Line 1: y = ⅓x + 2 Line 2: y = -x + 6
• Blue: Line 1 • Green: Line 2 • Red: (3, 3)
2
Problem 2: Elimination Method
\(\begin{cases} 2x + 3y = 18 \\ x - 2y = -5 \end{cases}\)
Solution Steps (Multiply Equation 2 by \(-2\)):
• Eq 1: \(2x + 3y = 18\)
• \(-2 \times\) Eq 2: \(-2x + 4y = 10\)
Add equations: \(7y = 28 \implies \mathbf{y = 4}\)
Back-substitute \(y = 4\) into Eq 2: \(x - 2(4) = -5 \implies x - 8 = -5 \implies \mathbf{x = 3}\)
Quick Check: \(2(3) + 3(4) = 6 + 12 = 18\) ✓
Solution:
\((x, y) = (3, 4)\)
Systems Practice Review • Answer Key Page 1 of 2
Answer Key • Problems 3 & 4 (Matrix Practice)
Teacher Solutions
3
Problem 3: Missing Variable Matrix & Solution
Zero in y-column
(a) Augmented Matrix (Note 0 in Row 2, Col 2):
\(\left[\begin{array}{ccc|c} 3 & 2 & -1 & 5 \\ 2 & \mathbf{0} & 3 & 13 \\ -1 & 4 & 1 & 5 \end{array}\right]\)
(b) Desmos Solution:
\(\mathbf{x = 2}, \quad \mathbf{y = 1}, \quad \mathbf{z = 3}\)
Ordered Triple: \((2, 1, 3)\)
4
Problem 4: Community Theater Ticket Modeling
Matrix Formulation & Solution
(a) Augmented Matrix in \(x, y, z\) Order:
Eq 1: \(x + y + z = 300\) • Eq 2: \(14x + 8y + 6z = 3100\) • Eq 3: \(0x + y - z = 20\)
\(\left[\begin{array}{ccc|c} 1 & 1 & 1 & 300 \\ 14 & 8 & 6 & 3100 \\ 0 & 1 & -1 & 20 \end{array}\right]\)
(b) Desmos rref Output & Tickets Sold:
\(\operatorname{rref}(A) \implies \begin{aligned} x &= 140 \\ y &= 90 \\ z &= 70 \end{aligned}\)
Adult (\(x\)): 140 tickets
Student (\(y\)): 90 tickets
Child (\(z\)): 70 tickets
Systems Practice Review • Answer Key Page 2 of 2
