Respuesta :
Step 1. To solve this problem we need to find a system of equations.
The variables we will use are:
[tex]\begin{gathered} x\longrightarrow\text{ pr}ice\text{ of 1 orange juice} \\ y\longrightarrow\text{price of 1 bagel} \\ z\longrightarrow\text{price of 1 cup of coffe}e \end{gathered}[/tex]originally, 1 orange juice, 1 bagel, and 1 cup of coffee cost $3.25, thus, we have the following equation:
[tex]x+y+z=3.25[/tex]Step 2. We can find a second equation considering the increases in the prices.
Now the orange juice costs 50% more, thus, instead of x, we will have 1.5x.
The bagels increased by 20%, thus, instead of y, we will have 1.2y. And the coffee stays at the same price "z".
Since the total cost after the increase was 4.05, the second equation is:
[tex]1.5x+1.2y+z=4.05[/tex]Step 3. So far, the system of equations is:
[tex]\begin{gathered} x+y+z=3.25 \\ 1.5x+1.2y+z=4.05 \end{gathered}[/tex]Step 4. Now we consider the relationship between the cost of orange juice and coffee.
After the increase, the orange juice (which now costs 1.5x) will cost twice as much as coffee.
This is represented in the following equation:
[tex]1.5x=2z[/tex]here, we can solve for z:
[tex]\begin{gathered} \frac{1.5x}{2}=z \\ 0.75x=z \end{gathered}[/tex]and substitute this value of z into our two equations from step 3:
[tex]\begin{gathered} x+y+0.75x=3.25 \\ 1.5x+1.2y+0.75x=4.05 \end{gathered}[/tex]Step 5. Simplify the equations by combining the terms containing x:
[tex]\begin{gathered} 1.75x+y=3.25 \\ 2.25x+1.2y=4.05 \end{gathered}[/tex]Step 6. We are going to solve this system of equations using the substitution method:
-solve for one of the variables in the first equation, and substitute that into the second equation.
-In this case, we solve for y in the first equation from step 5:
[tex]y=3.25-1.75x[/tex]And substitute this in the place of "y" of the second equation:
[tex]2.25x+1.2(3.25-1.75x)=4.05[/tex]Use the distributive property:
[tex]2.25+3.9-2.1x=4.05[/tex]And solve for x
-combine the like terms
[tex]6.15-2.1x=4.05[/tex]-subtract 6.15 to both sides:
[tex]\begin{gathered} -2.1x=4.05-6.15 \\ -2.1x=-2.1 \end{gathered}[/tex]-divide both sides of the equation by -2.1:
[tex]\begin{gathered} x=\frac{-2.1}{-2.1} \\ x=1 \end{gathered}[/tex]Step 7. we already have 1 of the values:
[tex]x=1\longrightarrow\text{ cost of 1 orange juice}[/tex]With this information and the relationship between x and z that we found in step 4:
[tex]0.75x=z[/tex]We can substitute x to find the value of z:
[tex]\begin{gathered} 0.75(1)=z \\ z=0.75\longrightarrow\cos t\text{ of 1 cup of coffe}e \end{gathered}[/tex]Step 8.
The last step is to substitute x into the first equation from step 6:
[tex]y=3.25-1.75x[/tex]Which relates y to x.
Substituting x=1:
[tex]\begin{gathered} y=3.25-1.75(1) \\ y=3.25-1.75 \\ y=1.5\longrightarrow\cos t\text{ of 1 bagel} \end{gathered}[/tex]Answer: Find the price of each item before the increase
Orange juice: $1
Bagel: $1.5
Coffee: $0.75
