Question

Mixture Asalt(table spoons) water(cups)4 57. 8 3/49. 11 1/4Mixture B is defined by the equation y=2.5x, where x is the number of teaspoons of salt and y is the number of cups of water.If you used 10 cups of water, which mixture would use more salt? How much more? Explain or show your reasoning.Which mixture tastes saltier? Explain how you know.

Answer 1

First, we must find the linear equation for mixture A. By definition, the linear equation is given by:

y = mx + b

where m is the slope of the line and b is the y-coordinate of the y-intercept. Now, by definition, the slope of the line is given by:

`m\text{ =}(Y1-Y2)/(X2-X1)\text{ }`

where (X1, Y1) and (X2,Y2) are points on the line (look at the table). In our case,

take for example:

(X1,Y1) = (4,5)

and

(X2,Y2) = (7,8)

Replacing these data in the equation of the slope we obtain:

`m\text{ =}(Y1-Y2)/(X2-X1)=(8-5)/(7-4)\text{ = }(3)/(3)\text{ = 1}`

Then, the slope of the line is m = 1. Now, to find b, we take any point on the table. For example (x,y) = (4,5) and we replace it in the equation of the line:

y = mx +b

or:

y = 1x + b

or

y = x + b

then b = y-x = 5-4 = 1

then we have that the equation for the Mixture A is:

`y\text{ =x + 1}`

On the other hand, by hypothesis, we have that the equation for Mixture B is:

`y\text{ =}2.5x`

Then we have that if we used 10 cups of water :

MIXTURE A:

10 = X + 1

then X = 10-1 = 9 teaspoons of salt

MIXTURE B:

`X\text{ = }(10)/(2.5)=\text{ 4}`

We can conclude that if we used 10 cups of water then the mixture that uses more salt would be: MIXTURE A