Q13. - Algebra 1

Question

The cost of two meals at a restaurant is shown in the table below.

Meal	Total Cost
$3ta\cos$, $2burritos$	$\mathrm{\backslash (}7.40$
$4ta\cos$, $1burritos$	$\mathrm{\backslash )}6.45$

a. Define variables to represent the cost of a taco and the cost of a burrito.

b. Write a system of equations to find the cost of a single taco and a single burrito.

c. Solve the systems of equations, and explain what the solution means.

d. How much would a customer pay for $2ta\cos$ and $2burritos$?

Step-by-Step Solution

Answer

a. The cost of one taco is $\mathrm{\$}1.08$ and the cost of one burrito is $\mathrm{\$}2.08$

b. The system of equations is $3x+2y=7.40$ and $4x+y=6.40$.

c. The solution of the system of equations is $\left(1.08,2.08\right)$.

d. The amount to be paid by a customer to buy 2 tacos and 2 burritos is $\mathrm{\$}6.32$.

Step 1. Define the concept.

1. Two equations of the form $a_{1}x+b_{1}y=c_1$ and $a_{2}x+b_{2}y=c_2$ form a system of equations. The ordered pair that is a solution of both equations is the solution of the system. A system can have one solution, an infinite number of solutions, or no solution.
2. Two equations of the form $a_{2}x+b_{2}y=c_2$ and $a_{2}x+b_{2}y=c_2$ form a system of equations. The ordered pair that is a solution of both equations is the solution of the system. A system can have one solution, an infinite number of solutions, or no solution.
3. The elimination method is the method in which the addition or subtraction method is used to get an equation in one variable.

     If the coefficients are opposite, use the addition and when the coefficients are the same, use the subtraction

Step 2. Assign the variables.

The cost of two meals is given which includes two items’ tacos and burritos.

Let the cost of being $x$ and cost of being $y$.

Step 3. Create the table to represent the assigned variables.

a. The table that represents the information of the variables is:

Tacos	Burritos	Cost (in dollars)
3	2	7.40
4	1	6.45

b.

The table that represents the information of the variables is:

Tacos	Burritos	Cost (in dollars)
3	2	7.40
4	1	6.45

In order-1, the total cost of $3ta\cos$ and $2burritos$ is $\mathrm{\$}7.40$. The cost of one taco is $x$ and that of the burrito is $y$. So the cost of $3ta\cos$ is $\mathrm{\$}3x$ and of $2burritos$ is width="28" style="max-width: none;" $\mathrm{\$}2y$. Therefore, the equation that connects the variable and given quantities is

$3x+2y=7.40$                            … (1)

In order-2, the total cost of $4ta\cos$ and $1burritos$ is width="40" style="max-width: none;" $\mathrm{\$}6.45$. The cost of one taco is $x$  and that of the burrito is $y$. So the cost of $3ta\cos$ is $\mathrm{\$}4x$ and of $2burritos$ is $\mathrm{\$}y$. Therefore, the equation that connects the variable and given quantities is 

$4x+y=6.40$                              … (2)

 

c.

 

The elimination method is the method in which the addition or subtraction method is used to get an equation in one variable. 

If the coefficients are opposite, use the addition and when the coefficients are the same, use the subtraction. 

 

The given system of equation is: 

$3x+2y=7.40$                            … (1)

$4x+y=6.40$                              … (2)

 

Multiply equation (2) by 2.

$$\begin{gathered} 2\left(4x+y\right)=2\times \left(6.40\right) \\ 8x+2y=12.8 \left(3\right) \end{gathered}$$

Note that both the equations (1) and (3) have  in same signs, therefore, the variable can be eliminated if both equations are subtracted.

$\begin{array}{l}3x+2y=7.4\\ \underset{¯}{\left(-\right)8x+2y=12.8}\\ -5x=-5.4\text{ Divide both sides by -5}\\ x=\text{1.08}\end{array}$

Substitute 1.08 $x$ into the equation (1) and solve for $y$

$$\begin{gathered} 3x+2y=7.40 \\ 3\left(1.08\right)+2y-\text{3.24}=7.40-\text{3.24} \\ 2y=\text{4.16} \\ y=\text{2.08} \end{gathered}$$

The solution is $\left(1.08,2.08\right)$.

 

d.

 

The table that represents the information of the variables is:

Tacos	Burritos	Cost (in dollars)
3	2	7.40
4	1	6.45

In order-1, the total cost of $3ta\cos$ and $2burritos$ is $\mathrm{\$}7.40$. The cost of one taco is $x$ and that of the burrito is $y$. So the cost of $3ta\cos$ is $\mathrm{\$}3x$ and of $2burritos$ is $\mathrm{\$}2y$. Therefore, the equation that connects the variable and given quantities is

$3x+2y=7.40$                            … (1)

In order-2, the total cost of $4ta\cos$ and $1burritos$ is $\mathrm{\$}6.45$. The cost of one taco is x and that of the burrito is y. So the cost of $3ta\cos$ is $\mathrm{\$}4x$ and of $2burritos$ is width="19" style="max-width: none;" $\mathrm{\$}y$. Therefore, the equation that connects the variable and given quantities is 

$4x+y=6.40$                              … (2)

 

After solving the system of equations using the elimination method, the solution is that the cost of $1taco$ is $\mathrm{\$}1.08$ and cost of $1burrito$ is $\mathrm{\$}2.08$.

 

The cost of one taco is $\mathrm{\$}1.08$ and the cost of one burrito is $\mathrm{\$}2.08$.

 

The amount paid by the customer to buy $2ta\cos$:

 

 $$\begin{gathered} c_1=2\times 1.08 \\ =2.16 \end{gathered}$$

 

The amount paid by the customer to buy :$2burritos$

 

$$\begin{gathered} c_2=2\times 2.08 \\ =4.16 \end{gathered}$$

 

Therefore, the amount to be paid by a customer to buy $2ta\cos$ and $2burrito$ is

 

 $$\begin{gathered} c_1+c_2=\mathrm{\$}2.16+\mathrm{\$}4.16 \\ =\mathrm{\$}\text{6.32} \end{gathered}$$

 

Thus, the amount to be paid by a customer to buy $2ta\cos$ and $2burrito$ is $\mathrm{\$}6.32$.