Question

john estimates the value of his car over time. the equation for the line of best fit is approximated as ( y = - 2.9 x + 15.7 ) where ( y ) represents the value, in thousands of dollars. what values complete the residual table? ( a = ) ( b = ) ( c = ) ( d = )

Explanation:

Step1: Recall Residual Formula

Residual = Given Value - Predicted Value. Also, Predicted Value is from \( y = -2.9x + 15.7 \), where \( x \) is age (years).

Step2: Find \( a \) (Predicted Value when \( x = 1 \))

Use the line of best fit: \( y = -2.9(1) + 15.7 \)
\( y = -2.9 + 15.7 = 12.8 \)? Wait, no—wait, residual is Given - Predicted. Wait, residual for \( x = 1 \) is \( 15 - a = 0.2 \)? Wait, no, wait the table: Age 1, Given Value 15, Residual 0.2. So Residual = Given - Predicted → \( 15 - a = 0.2 \) → \( a = 15 - 0.2 = 14.8 \). Wait, but also check the line of best fit. Wait, maybe I mixed up. Wait, the line of best fit is \( y = -2.9x + 15.7 \). For \( x = 1 \), \( y = -2.9(1) + 15.7 = 12.8 \)? But that contradicts the residual. Wait, no, maybe the residual is Predicted - Given? No, residual is usually (Observed - Predicted). Let's check the last row: Age 5, Given 4, Predicted 3.2, Residual 0.8. 4 - 3.2 = 0.8. Yes! So Residual = Given Value - Predicted Value. So for Age 1: Given 15, Residual 0.2 → 15 - a = 0.2 → a = 15 - 0.2 = 14.8? Wait, but let's check the line of best fit. Wait, maybe the line of best fit is \( y = -2.9x + 15.7 \). Wait, for \( x = 1 \), \( y = -2.9(1) + 15.7 = 12.8 \). But that's not 14.8. Wait, maybe a typo? Wait, no, the last row: Age 5, Predicted 3.2. Let's check the line: \( y = -2.9(5) + 15.7 = -14.5 + 15.7 = 1.2 \)? Wait, no, 3.2? Wait, maybe the equation is \( y = -2.9x + 15.7 \)? Wait, 5*(-2.9) = -14.5, 15.7 -14.5 = 1.2. But the table says Predicted Value for Age 5 is 3.2. Wait, maybe the equation is \( y = -2.9x + 15.7 \)? Wait, no, maybe the equation is \( y = -2.9x + 15.7 \)? Wait, let's re-express. Wait, the problem says "the equation for the line of best fit is approximated as \( y = -2.9x + 15.7 \), where \( y \) represents the value, in thousands of dollars." Wait, for Age 5 (x=5), y should be -2.9*5 +15.7 = -14.5 +15.7 = 1.2. But the table has Predicted Value 3.2. So there's a mistake? Wait, no, maybe the equation is \( y = -2.9x + 15.7 \)? Wait, no, maybe I misread the equation. Wait, maybe it's \( y = -2.9x + 15.7 \)? Wait, let's check Age 2: Predicted Value 11.9. Let's plug x=2: \( y = -2.9(2) +15.7 = -5.8 +15.7 = 9.9 \). But the table says Predicted Value 11.9. So clearly, the equation must be \( y = -2.9x + 15.7 \)? No, that's not matching. Wait, maybe the equation is \( y = -2.9x + 15.7 \)? Wait, no, maybe the equation is \( y = -2.9x + 15.7 \)? Wait, let's check the last row: Age 5, Predicted 3.2. Let's solve for the equation. If x=5, y=3.2: 3.2 = -2.9*5 + b → 3.2 = -14.5 + b → b=17.7. Ah! So the equation is \( y = -2.9x + 17.7 \). Yes! That makes sense. Because -2.9*5 +17.7 = -14.5 +17.7 = 3.2. Yes! So the equation is \( y = -2.9x + 17.7 \). Okay, that's the correct equation. So let's redo:

Step1: Find \( a \) (Predicted Value when \( x=1 \))

Use \( y = -2.9x + 17.7 \). For \( x=1 \): \( y = -2.9(1) +17.7 = 14.8 \). So \( a = 14.8 \). Let's check residual: Given Value (15) - Predicted Value (14.8) = 0.2. Which matches the table. Good.

Step2: Find \( b \) (Residual for \( x=2 \))

Given Value =12, Predicted Value=11.9 (from table). Residual = Given - Predicted = 12 - 11.9 = 0.1. Wait, but let's check with the equation. For \( x=2 \), Predicted Value is \( y = -2.9(2) +17.7 = -5.8 +17.7 = 11.9 \). So Residual = 12 - 11.9 = 0.1. So \( b = 0.1 \).

Step3: Find \( c \) (Predicted Value when \( x=3 \))

Residual = 0 (Given Value - Predicted Value =0 → Given Value = Predicted Value). Given Value is 9, so Predicted Value \( c = 9 \). Let's check with the equation: \( y = -2.9(3) +17.7 = -8.7 +17.7 = 9 \). Perfect.

Step4: Find \( d \) (Residual for \( x=4 \))

Given Value=5, Predicted Value=6.1 (from table). Residual = Given - Predicted = 5 - 6.1 = -1.1. Let's check with the equation: \( y = -2.9(4) +17.7 = -11.6 +17.7 = 6.1 \). So Residual = 5 - 6.1 = -1.1. So \( d = -1.1 \).

Answer:

\( a = 14.8 \)
\( b = 0.1 \)
\( c = 9 \)
\( d = -1.1 \)