Respuesta :
ANSWER
• Apple = $0.29
,• Banana = $0.05
,• Cantaloupe = $0.89
EXPLANATION
First, we have to name the variables:
• x = cost of one apple
,• y = cost of one banana
,• z = cost of one cantaloupe
Next, we have to write equations for the purchases each person did:
• Liz,: 3 apples (3x), 12 bananas (12y), and 1 cantaloupe (z) for $2.36
,• Bob,: 12 apples (12x), no bananas (0y), and 2 cantaloupe (2z) for $5.26
,• Carol,: no apples (0x), 2 bananas (2y), and 3 cantaloupe (3z) for $2.77
We have the following system of equations,
[tex]\begin{cases}3x+12y+z=2.36 \\ 12x+2z=5.26 \\ 2y+3z=2.77\end{cases}[/tex]We can solve this system using the method of substitution.
Solve the second equation for x,
[tex]\begin{gathered} 12x+2z=5.26 \\ \\ 12x=5.26-2z \\ \\ x=\frac{5.26}{12}-\frac{2z}{12} \end{gathered}[/tex]Solve the third equation for y,
[tex]\begin{gathered} 2y+3z=2.77 \\ \\ 2y=2.77-3z \\ \\ y=\frac{2.77}{2}-\frac{3z}{2} \end{gathered}[/tex]Next, replace x and y with these two expressions as functions of z in the first equation,
[tex]3\mleft(\frac{5.26}{12}-\frac{2z}{12}\mright)+12\mleft(\frac{2.77}{2}-\frac{3z}{2}\mright)+z=2.36[/tex]We have an equation for z. Apply the distributive property to eliminate the parenthesis,
[tex]3\cdot\frac{5.26}{12}-3\cdot\frac{2z}{12}+12\cdot\frac{2.77}{2}-12\cdot\frac{3z}{2}+z=2.36[/tex]Solve the products and quotients - with the help of a calculator,
[tex]1.315-0.5z+16.62-18z+z=2.36[/tex]Add like terms,
[tex]\begin{gathered} (1.315+16.62)+(-0.5z-18z+z)=2.36 \\ \\ 17.935-17.5z=2.36 \end{gathered}[/tex]Subtract 17.935 from both sides,
[tex]\begin{gathered} 17.935-17.935-17.5z=2.36-17.935 \\ \\ -17.5z=-15.575 \end{gathered}[/tex]And divide both sides by -17.5,
[tex]\begin{gathered} \frac{-17.5z}{-17.5}=\frac{-15.575}{-17.5} \\ \\ z=0.89 \end{gathered}[/tex]Now, knowing that z = 0.89, we can replace this value into the second and third equations that we solved for x and y before,
[tex]x=\frac{5.26}{12}-\frac{2z}{12}=\frac{5.26}{12}-\frac{2\cdot0.89}{12}[/tex]Solving this with the help of a calculator, we get x = 0.29.
For y,
[tex]y=\frac{2.77}{2}-\frac{3z}{2}=y=\frac{2.77}{2}-\frac{3\cdot0.89}{2}[/tex]Again, using a calculator, we have that y = 0.05.
Hence, the cost of each fruit is:
• Apple = $0.29
,• Banana = $0.05
,• Cantaloupe = $0.89
