Modeling Linear Relationships
Aligned to 8.F.B.4 — Common Core State Standards for Mathematics.
What this lesson teaches
To model a linear relationship, find the rate of change (slope) and the initial value (y-intercept), then write y = mx + b. From two points, slope = (change in y) / (change in x).
Worked examples
Example 1
A plumber charges a $40 fee plus $30 per hour. The initial value is b = 40 and the rate is m = 30, so y = 30x + 40. For 3 hours: y = 30(3) + 40 = $130.
Example 2
A tank holds 500 gallons and drains 25 per hour: y = −25x + 500. It is empty when y = 0, at x = 20 hours.
Example 3
A negative slope means decreasing. If the rate in a story is 'loses', 'drains', or 'falls', the slope should come out negative.
Practice questions
- Write an equation: a gym costs $25 to join plus $15 per month.
- Find the equation of the line through (1, 7) and (3, 15).
- A candle is 20 cm tall and burns 4 cm per hour. Write its equation and find its height after 3 hours.
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
Modelado de Relaciones Lineales
Para modelar una relación lineal, halla la tasa de cambio (pendiente) y el valor inicial (intersección con el eje y), y luego escribe y = mx + b. A partir de dos puntos, pendiente = (cambio en y) / (cambio en x).
Ejemplo 1: Un plomero cobra una tarifa de $40 más $30 por hora. El valor inicial es b = 40 y la tasa es m = 30, así que y = 30x + 40. Para 3 horas: y = 30(3) + 40 = $130.
Ejemplo 2: Un tanque tiene 500 galones y se vacía 25 por hora: y = −25x + 500. Queda vacío cuando y = 0, a las x = 20 horas.
Ejemplo 3: Una pendiente negativa significa que disminuye. Si la tasa en el problema es 'pierde', 'se vacía' o 'baja', la pendiente debe salir negativa.
More Functions lessons
Teach this lesson today
An AI tutor that guides with questions instead of just giving answers — all K–12 core subjects, every child in the family.
Start free — no card required