Introduce variables for the amount of blue and red food coloring needed.
19ml blue and 6ml red
Practice makes perfect
Let's introduce variables for the amount of blue and red food coloring Peggy needs to make purple.
Description
Variable
Blue food coloring
bml
Red food coloring
rml
Let's use these variables to write equations expressing the information given in the question.
Description
Equation
Peggy needs 25ml food coloring.
b+r=25
The amount of food coloring in bml 50 % concentration blue food coloring.
0.50bml
The amount of food coloring in rml 25 % concentration red food coloring.
0.25rml
The amount of food coloring in 25ml 44 % concentration mix.
0.44* 25=11ml
The coloring in the mix comes from the blue and red concentration.
0.50b+0.25r=11
We now have two equations involving the two unknowns, b and r.
Let's write this system of equations as a matrix equation.
b+r=25 0.50b+0.25r=11
⇒
b+r
0.50b+0.25r
=
25 11
Now we will write the left-hand side of the above matrix equation as the product of the coefficient matrix and the variable matrix.
b+r 0.50b+0.25r
=
25 11
⇕
1& 1 0.50& 0.25
*
b r
=
25 11
We will solve the matrix equation by multiplying both sides by the inverse of the coefficient matrix to isolate the b and r variables. When multiplying a matrix by its inverse, we obtain the identity matrix.
