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Difficulty: MediumLinear Equations in Two Variables

An online retail company determines that the relationship between the selling price of a product, xx dollars, and the daily number of units sold, yy, can be modeled by a linear equation. When the selling price is 1212 dollars, the company sells 8080 units per day. For every 33 dollars increase in the selling price, the number of units sold daily decreases by 1515. Which of the following equations represents this relationship?

  1. y=−5x+140y = -5x + 140Answer
  2. B
    y=−5x+80y = -5x + 80
  3. C
    y=−0.2x+82.4y = -0.2x + 82.4
  4. D
    y=−5x+20y = -5x + 20

Answer

The equation y=−5x+140y = -5x + 140
The relationship between price xx and units sold yy is linear. The slope mm represents the rate of change: m=ΔyΔx=−153=−5m = \frac{\Delta y}{\Delta x} = \frac{-15}{3} = -5. Using the point-slope form with the known point (12,80)(12, 80), we get y−80=−5(x−12)y - 80 = -5(x - 12). Distributing the −5-5 gives y−80=−5x+60y - 80 = -5x + 60. Adding 8080 to both sides results in the equation y=−5x+140y = -5x + 140.

Step-by-Step Solution

1
Calculate the slope (mm) of the linear relation.
m=change in ychange in x=−153=−5m = \frac{\text{change in } y}{\text{change in } x} = \frac{-15}{3} = -5
The slope is the rate of change, which represents the decrease of 1515 units for every 33 dollars increase in price.
2
Set up the equation using point-slope form with the point (12,80)(12, 80) and the slope m=−5m = -5.
y−80=−5(x−12)y - 80 = -5(x - 12)
Point-slope form allows us to write the equation of a line given its slope and a point it passes through.
3
Simplify the equation into slope-intercept form.
y−80=−5x+60⇒y=−5x+140y - 80 = -5x + 60 \Rightarrow y = -5x + 140
Distributing the slope −5-5 to −12-12 yields +60+60, and adding 8080 to both sides isolates the variable yy.

Key Concept

Linear Equations in Two Variables

Alternative Method

Substitute the point (12,80)(12, 80) and slope m=−5m = -5 into the slope-intercept equation y=mx+by = mx + b to find bb: 80=−5(12)+b⇒80=−60+b⇒b=14080 = -5(12) + b \Rightarrow 80 = -60 + b \Rightarrow b = 140. Thus, y=−5x+140y = -5x + 140.
Estimated Time:1m 30s
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Topics

Linear Equations in Two Variables

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