The linear equation represented on the coordinate plane represents the average cost of a loaf of white bread since the year 1985.

Question

(10, 0.85) (17, 1.06) (23, 1.24) (30, 1.45)
Select all the statements that are true for the linear equation.

(4 points)
Responses

y = 0.03x + 0.55 represents the linear equation
y = 0.03x + 0.55 represents the linear equation

The cost of a loaf of bread in 1990 was $0.70.
The cost of a loaf of bread in 1990 was $0.70.

(27,1.48) is a solution to the linear equation
(27,1.48) is a solution to the linear equation

The cost of a loaf of bread in 2015 was $1.50
The cost of a loaf of bread in 2015 was $1.50

(5, 0.7) is a solution to the linear equation
(5, 0.7) is a solution to the linear equation

A loaf of bread cost $0.55 in 1985

Answer 1

To determine which statements are true regarding the linear equation representing the average cost of a loaf of white bread since 1985, we need to analyze the given coordinates and the proposed linear equation.

1. Finding the linear equation: Given points are:

   • (10, 0.85), which represents the year 1995 (10 years after 1985)
   • (17, 1.06), which represents the year 2002 (17 years after 1985)
   • (23, 1.24), which represents the year 2008 (23 years after 1985)
   • (30, 1.45), which represents the year 2015 (30 years after 1985)

   To find the slope (m) of the line connecting any two points, we can use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Let's take the points (10, 0.85) and (30, 1.45): \[ m = \frac{1.45 - 0.85}{30 - 10} = \frac{0.60}{20} = 0.03 \]

   Now we can find the y-intercept (b) using the equation of a line \(y = mx + b\). Using the point (10, 0.85): \[ 0.85 = 0.03(10) + b \implies 0.85 = 0.30 + b \implies b = 0.55 \] Therefore, the equation of the line is: \[ y = 0.03x + 0.55 \]

2. Evaluating each statement:

   • y = 0.03x + 0.55 represents the linear equation.

     • True (confirmed by our calculation)

   • The cost of a loaf of bread in 1990 was $0.70.

     • 1990 corresponds to \(x = 5\): \[ y = 0.03(5) + 0.55 = 0.15 + 0.55 = 0.70 \]
     • True

   • (27, 1.48) is a solution to the linear equation.

     • Plugging \(x = 27\): \[ y = 0.03(27) + 0.55 = 0.81 + 0.55 = 1.36 \]
     • Since \(1.48 \neq 1.36\), this statement is False.

   • The cost of a loaf of bread in 2015 was $1.50.

     • 2015 corresponds to \(x = 30\): \[ y = 0.03(30) + 0.55 = 0.90 + 0.55 = 1.45 \]
     • Since $1.50 does not equal $1.45, this statement is False.

   • (5, 0.7) is a solution to the linear equation.

     • Plugging \(x = 5\): \[ y = 0.03(5) + 0.55 = 0.70 \]
     • True

   • A loaf of bread cost $0.55 in 1985.

     • The year 1985 corresponds to \(x = 0\): \[ y = 0.03(0) + 0.55 = 0.55 \]
     • True

In conclusion, the true statements are:

1. y = 0.03x + 0.55 represents the linear equation.
2. The cost of a loaf of bread in 1990 was $0.70.
3. (5, 0.7) is a solution to the linear equation.
4. A loaf of bread cost $0.55 in 1985.