Question

At a farmers' market. Karen and Alice each bought some bread that cost $2 per loaf. Then Karen spent $3 to purchase other items, while Alice spent $1. Could the girls have bought the same number of loaves of bread and spent the same total amount?

1. Let x represent the number of loaves of bread. Fill in the boxes to write a system of equations for the total amounts spent.

Karen: y = x +
Alice: y = x +

2. What are the slopes of the lines represented by the equations?

3. What are the y-intercepts of the lines represented by the equations?

4. Do the lines intersect?

5. How many solutions are there to this system of equations?

math6. Can Karen and Alice have bought the same number of loaves of bread and spent the same total amount?​

Answer 1

(1)

Karen:
`y = 2x +3`

Alice:
`y = 2x +1`

(2): Slope is 2

(3) y intercept

`b = 3` --- For Karen

`b = 1` --- For Alice

(4) They do not intersect

(5) No solution

(6) No

Explanation:

Given

`x = loaves`

`y = amount`

Solving (1): Complete the equation.

For Karen:

`m = 2` --- the rate

`b = 3` --- additional spendings

For Alice:

`m = 2` --- the rate

`b = 1` --- additional spendings

The general equation is;

`y = mx +b`

For Karen, it becomes

`y = 2x +3`

For Alice, it is:

`y = 2x +1`

(2): The slopes.

In (1), we have:

`m = 2` --- the rate (for both Karen and Alice).

This rate implies the slope.

Hence, the slope is 2

(3): The y intercept.

The additional spendings in (1) means the y intercept.

So, we have:

`b = 3` --- For Karen

`b = 1` --- For Alice

(4) Because they have the same slope, they do not intersect

(5) Number of solutions.

`y = 2x +3`

`y = 2x +1`

Subtract both equations

`y-y = 2x-2x +3-1`

`0 = 2`

This means there is no solution

(6) Can they spend the same amount on the same number of loaves

In (5), the equations have no solution

Hence, it is not possible to spend the same amount for the same number of loaves