Creating Equations to Model Problems
Aligned to A-CED.A.1 — Common Core State Standards for Mathematics.
What this lesson teaches
Translate a real situation into an equation or inequality by naming the unknown, then writing how the quantities relate. Solving the equation answers the question the situation asks.
Worked examples
Example 1
A gym charges a $50 signup fee plus $30 per month. When does the total cost reach $200? Model it as 50 + 30m = 200. Subtract 50: 30m = 150. Divide by 30: m = 5 months.
Example 2
A rectangular garden must have a perimeter of 60 feet and an area of at least 200 square feet. Let w be the width. Perimeter gives 2w + 2L = 60, so L = 30 − w. The area condition becomes w(30 − w) ≥ 200, or −w² + 30w − 200 ≥ 0. Testing w = 10: 10(20) = 200 ✓, so a 10-by-20 garden is the smallest width that works.
Example 3
Common mistake: writing the equation without naming what the variable means. An answer of x = 12 is useless if nobody recorded whether x was months, dollars, or people. Start every model with a sentence — let m = the number of months — and end by translating the number back: m = 5 means five months. Check the units on both sides too: 50 + 30m = 200 only balances if 30 is dollars PER MONTH.
Practice questions
- A phone plan costs $20 plus $0.10 per text. Write an equation for a $35 bill.
- You save $15 each week toward a $180 goal. Write and solve an equation for the number of weeks.
- Write an inequality: a car rental is $40 plus $0.25 per mile and must stay under $100.
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Crear Ecuaciones para Modelar Problemas
Traduce una situación real a una ecuación o desigualdad nombrando la incógnita y luego escribiendo cómo se relacionan las cantidades. Resolver la ecuación responde lo que la situación pregunta.
Ejemplo 1: Un gimnasio cobra una inscripción de $50 más $30 por mes. ¿Cuándo llega el costo total a $200? Modélalo como 50 + 30m = 200. Resta 50: 30m = 150. Divide entre 30: m = 5 meses.
Ejemplo 2: Un jardín rectangular debe tener 60 pies de perímetro y al menos 200 pies cuadrados de área. Sea w el ancho. El perímetro da 2w + 2L = 60, así que L = 30 − w. La condición de área queda w(30 − w) ≥ 200, o −w² + 30w − 200 ≥ 0. Probando w = 10: 10(20) = 200 ✓, así que un jardín de 10 por 20 es el ancho mínimo que sirve.
Ejemplo 3: Error común: escribir la ecuación sin decir qué significa la variable. Una respuesta de x = 12 no sirve si nadie anotó si x eran meses, dólares o personas. Empieza cada modelo con una frase — sea m = el número de meses — y termina traduciendo el número: m = 5 significa cinco meses. Revisa también las unidades de ambos lados: 50 + 30m = 200 solo cuadra si 30 son dólares POR MES.
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