Q28. - Algebra 2

Question

Maka loves the lunch combinations at Rosita’s Mexican Restaurant. Today however, she wants a different combination than the ones listed on the menu.

 

If Maka wants 2 and 1 enchilada, how much should she plan to spend.

Step-by-Step Solution

Answer

The total amount that Maka needs to spent on 2 burritos and one enchilada is$7.80$.

Step 1 – First construct a system of equations corresponding to the given situation.

Let $x$ be the price of an enchilada, $y$ be the price of a taco, and $z$be the price of a burrito.

 In the figure, it is given that the total cost of two tacos, and one burrito is %6.55.

So, $2y+z=6.55$.

 In the figure, it is given that the total cost of one enchilada, one taco, and one burrito is %7.10.

So, $x+y+z=7.10$

 In the figure, it is given that the total cost of 2 tacos and two enchiladas is %8.90.

This can be mathematically written as $2x+2y=8.90$.

 So, the system of equations will be:

$\begin{array}{c}2y+z=6.55\\ x+y+z=7.10\\ 2x+2y=8.90\end{array}$

Step 2 – Use the elimination method to get the system of equations in two variables.

Multiply the $equation x+y+z=7.10$ by 2 and subtract the new resultant equation to the equation $2y+z=6.55$.

$\begin{array}{l}\underset{\_}{\begin{array}{l}x+y+z=7.10\\ 2y+z=6.55\end{array}}\begin{array}{c}\text{multiply by 2}\\ \end{array}\underset{\_}{\begin{array}{l}2x+2y+2z=14.20\\ (-)+2y+z=6.55\end{array}}\\ 2x+0+z=7.65\end{array}$

So, the resultant equation is $2x+z=7.65$.

 Multiply the $equation x+y+z=7.10$ by 2 and subtract the new resultant equation from the equation $2x+2y=8.90$

$\begin{array}{l}\underset{\_}{\begin{array}{l}x+y+z=7.10\\ 2x+2y=8.90\end{array}}\begin{array}{c}\text{multiply by 2}\\ \end{array}\underset{\_}{\begin{array}{l}2x+2y+2z=14.20\\ (-)2x+2y+=8.90\end{array}}\\ 0+0+2z=5.30\end{array}$

So, the resultant equation is $2z =5.30$.

Step 3 – Solve the system of equations in two variables

Solve $2z =5.30$ for $z$:

$\begin{array}{c}2z=5.30\\ \frac{2z}{2}=\frac{5.30}{2}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} divide both sides by }2\\ z=2.65\end{array}$

Step 4 – Find the values of x and y .

Substitute $z=2.65$ in $2x+z=7.65$ and find the value of $x$.

$\begin{array}{c}2x+z=7.65\\ 2x+2.65=7.65\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Substitute }2.65\text{ for }z\\ 2x=5\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Subtract }2.65\text{ from both sides}\\ x=2.5\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Divide both sides by 2}\end{array}$

Substitute $z=2.65$ in $2y+z=6.55$ and find the value of $y$

$\begin{array}{c}2y+z=6.55\\ 2y+2.65=6.55\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Substitute }2.65\text{ for }z\\ 2y=3.90\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Subtract }2.65\text{ from both sides}\\ y=1.95\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Divide both sides by 2}\end{array}$

Hence, the solution of the given system of equations is$\left(x,y,z\right)=\left(2.5,1.95,2.65\right)$.

.

Step 5 – Use the substitution and find the total amount that spent on 2 burritos and one enchilada.

The mathematical expression that gives the total amount of 2 burritos and one enchilada is $2z+x.$

Substitute$x=2.5,z=2.65$ in $2z+x.$

$\begin{array}{c}2z+x=2\left(2.65\right)+2.5\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} substitute }2.65\text{ for }z,2.5\text{ for }x\\ =5.30+2.5\text{ simplify}\\ =7.80\end{array}$

Hence, the total amount that Maka needs to spent on 2 burritos and one enchilada is $7.80$.