Algebraic Modeling

Translate real situations into algebraic models and interpret results.

Learning objectives

After this lesson you should be able to define variables with units, translate words into equations, choose a model family, solve for unknowns, interpret results, and use SymPy to check model parameters.

Motivation: a model is a useful simplification

A model is a simplified mathematical description. A road map is not the city, but it helps you travel. An algebraic model is not the full situation, but it helps you calculate, compare, predict, and decide.

The modeling cycle

  1. Understand the situation.
  2. Define variables with units.
  3. State assumptions.
  4. Choose a model family.
  5. Translate relationships into equations.
  6. Solve or analyze.
  7. Interpret the result.
  8. Check reasonableness and limitations.

Key idea

Choosing variables and interpreting answers are part of the mathematics.

Variables, parameters, and units

In C(n) = 50 + 12n, n might be number of items, C(n) cost in dollars, 50 a fixed cost, and 12 dollars per item. Units help you catch mistakes.

Translating words into algebra

Words Algebra
five more than x x + 5
five less than x x - 5
twice x 2x
x decreased by 20% 0.80x
total adult and student tickets a + s
revenue from adult tickets at $12 each 12a

Common mistake

Do not start solving before defining what each variable means.

Linear models

Linear models have the form y = mx + b. The slope is a constant rate of change, and the intercept is the starting value.

A repair service charges $40 to visit and $25 per hour:

C(h) = 40 + 25h

A plant is 10 cm tall on day 0 and 22 cm tall on day 6:

slope = (22 - 10)/(6 - 0) = 2 cm/day
H(t) = 10 + 2t

\(\displaystyle 2 x + 10\)

Systems as models

A club sold 50 tickets. Adult tickets cost $10, student tickets cost $6, and total revenue was $380.

a + s = 50
10a + 6s = 380

Substitute s = 50 - a:

10a + 6(50 - a) = 380
4a = 80
a = 20
s = 30

Quadratic models

Quadratic models describe curved relationships and often have a maximum or minimum. A height model such as h(t) = -16t^2 + 64t + 5 opens downward, so its vertex is the maximum height.

A vertex form model is:

f(x) = a(x - h)^2 + k

If the vertex is (3, 10) and the graph passes through (5, 2), then 2 = a(5 - 3)^2 + 10, so a = -2.

Rational, radical, exponential, and logarithmic models

  • Rational: time for fixed distance, T(r) = 120/r, with r > 0.
  • Radical: side length from area, s(A) = sqrt(A), with A >= 0.
  • Exponential: repeated percentage growth, P(t)=P_0(1+r)^t.
  • Logarithmic: solving an exponential model for time.

Model checking

Ask whether units are consistent, the answer is in the domain, the size and sign are reasonable, whole numbers are required, and the model is being used outside its intended range.

Browser symbolic practice

Use math.js for browser calculators, Nerdamer for parameter solving, Algebrite for CAS exploration, or PyScript for a heavier Python/SymPy sandbox.

Try entering two points to compute a line, evaluating a candidate formula at data points, and comparing linear and exponential predictions.

Practice exercises

  1. Translate: “seven more than twice x.”
  2. A taxi charges $4 plus $2.50 per mile. Write a cost model.
  3. A line passes through (1, 5) and (4, 17). Find a linear model.
  4. A store sells notebooks for $3 and pens for $2. A customer buys 12 items for $31. Set up and solve a system.
  5. A square has area A. Write side length as a function of A.
  6. A value starts at 800 and decays by 6% per year. Write a model.
  7. A quadratic has vertex (2, 9) and passes through (4, 1). Find a.
  8. Explain one limitation of a taxi cost model.
  1. 2x + 7.
  2. C(m)=4+2.50m.
  3. y = 4x + 1.
  4. n+p=12, 3n+2p=31; n=7, p=5.
  5. s(A)=sqrt(A), A >= 0.
  6. V(t)=800(0.94)^t.
  7. 1=a(4-2)^2+9, so a=-2.
  8. It may ignore minimum fares, traffic charges, rounding, or rate changes.

Subtopic guided practice and checkpoints

Modeling is not just solving. It is choosing variables, translating conditions, solving, and checking whether the result makes sense in context.

Lab 1: define variables and units

Guess first. If a problem asks for time and distance, why is it dangerous to write equations before defining variables?

Guided exercise.

Suppose a taxi charges $4 plus $2.50 per mile. Define:

m = number of miles
C = total cost in dollars

Then the model is:

C = 4 + 2.50m

Checkpoint. A gym charges a $25 sign-up fee plus $18 per month. Define variables with units and write a cost model.

Lab 2: translate words into equations

Guess first. In “five more than twice a number is seventeen,” which part is the variable expression?

Guided exercise.

number -> n
twice a number -> 2n
five more than twice a number -> 2n + 5
is seventeen -> = 17

So:

2n + 5 = 17
2n = 12
n = 6

Checkpoint. Translate and solve: “Three less than four times a number is twenty-one.”

Lab 3: linear model from two data points

Guess first. If data points are (2, 11) and (6, 23), is the rate of change positive or negative?

Guided exercise.

m = (23 - 11)/(6 - 2) = 12/4 = 3
y - 11 = 3(x - 2)
y = 3x + 5

The model predicts an output of 5 when x = 0.

Checkpoint. Build a linear model through (1, 9) and (5, 21). Interpret the slope.

Lab 4: check a model result

Guess first. If a ticket problem gives a = -3 adult tickets, should you accept the algebraic answer?

Guided exercise.

A model answer must satisfy three tests:

  1. It satisfies the equation or system.
  2. It has the right units.
  3. It makes sense in the context.

Negative ticket counts, negative lengths, and fractional people usually mean the model or interpretation needs review.

Checkpoint. A calculation gives 2.4 buses for a field trip. What final answer should the planner use, and why?

Guessing game checkpoint

Using this lesson with edumath and SymPy

Use parsed equations to check the algebra after you have defined variables and units. The computer can solve, but you must decide whether the model makes sense.

from edumath.core import parse_equation
from edumath.solvers import solve_equation_steps

model_solution = solve_equation_steps(parse_equation("2*n + 5 = 17"))
print(model_solution.render_text())
Answer: n = 6

Method: linear equation

Steps:

1. Original equation:
Eq(2*n + 5, 17)

2. Move everything to one side:
Eq(2*n - 12, 0)
This writes the equation in the form ax + b = 0.

3. Move the constant term:
Eq(2*n, 12)

4. Divide by the coefficient of the variable:
Eq(n, 6)

Check:

n = 6: valid
import sympy as sp

m, b = sp.symbols("m b")
sp.solve((sp.Eq(m*2 + b, 7), sp.Eq(m*5 + b, 16)), (m, b))
{b: 1, m: 3}
a, x = sp.symbols("a x")
model = a*(x - 3)**2 + 10
sp.solve(sp.Eq(model.subs(x, 5), 2), a)
[-2]
from edumath.algebra import expression_table

expression_table("35 + 8*x", inputs=(0, 1, 2, 3, 4))
((0.0, 35.0), (1.0, 43.0), (2.0, 51.0), (3.0, 59.0), (4.0, 67.0))

Further reading