Let's solve this step-by-step.
### Step A: Write a system of equations
Given:
1. Two burgers and three orders of fries cost \[tex]$19.75.
2. Five burgers and two orders of fries cost \$[/tex]37.
Let [tex]\( x \)[/tex] be the cost of one burger.
Let [tex]\( y \)[/tex] be the cost of one order of fries.
We can translate the information into a system of linear equations:
1. [tex]\( 2x + 3y = 19.75 \)[/tex]
2. [tex]\( 5x + 2y = 37 \)[/tex]
### Step B: Solve the problem
Now, we need to solve this system of equations to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
#### Step 1: Multiply the equations to align the coefficients
To eliminate one of the variables, we can align the coefficients of either [tex]\( x \)[/tex] or [tex]\( y \)[/tex]. Let's eliminate [tex]\( y \)[/tex].
We'll multiply the first equation by 2 and the second equation by 3 to get the coefficients of [tex]\( y \)[/tex] to match:
[tex]\[ 2(2x + 3y) = 2 \times 19.75 \][/tex]
[tex]\[ 3(5x + 2y) = 3 \times 37 \][/tex]
This gives us:
[tex]\[ 4x + 6y = 39.50 \][/tex]
[tex]\[ 15x + 6y = 111 \][/tex]
#### Step 2: Subtract the equations to eliminate [tex]\( y \)[/tex]
Now subtract the first modified equation from the second modified equation:
[tex]\[ (15x + 6y) - (4x + 6y) = 111 - 39.50 \][/tex]
[tex]\[ 11x = 71.50 \][/tex]
#### Step 3: Solve for [tex]\( x \)[/tex]
[tex]\[ x = \frac{71.50}{11} = 6.50 \][/tex]
So, the cost of one burger is [tex]\( \$6.50 \)[/tex].
#### Step 4: Substitute [tex]\( x \)[/tex] back into one of the original equations to solve for [tex]\( y \)[/tex]
Using the first original equation:
[tex]\[ 2x + 3y = 19.75 \][/tex]
[tex]\[ 2(6.50) + 3y = 19.75 \][/tex]
[tex]\[ 13 + 3y = 19.75 \][/tex]
[tex]\[ 3y = 19.75 - 13 \][/tex]
[tex]\[ 3y = 6.75 \][/tex]
[tex]\[ y = \frac{6.75}{3} = 2.25 \][/tex]
So, the cost of one order of fries is [tex]\( \$2.25 \)[/tex].
### Step C: Solution to the system of equations
The cost of one burger is \[tex]$6.50, and the cost of one order of fries is \$[/tex]2.25.
