Let's denote:
- [tex]\( x \)[/tex] as the price per gallon of midgrade gas.
- [tex]\( y \)[/tex] as the price per gallon of regular gas.
We have two scenarios with two sets of data, which give us two equations to work with.
### Week 1
Tom bought:
- 18 gallons of midgrade gas.
- 5 gallons of regular gas.
He spent a total of \[tex]$59.91.
This can be represented with the following equation:
\[ 18x + 5y = 59.91 \]
### Week 2
Tom bought:
- 14 gallons of midgrade gas.
- 1 gallon of regular gas.
He spent a total of \$[/tex]397.
This can be represented with the following equation:
[tex]\[ 14x + y = 397 \][/tex]
### Solving the System of Equations
We have the following system of linear equations:
1. [tex]\( 18x + 5y = 59.91 \)[/tex]
2. [tex]\( 14x + y = 397 \)[/tex]
To solve for [tex]\( x \)[/tex] and [tex]\( y \)[/tex], we can use methods such as substitution, elimination, or matrix operations, but here, we will solve them directly to find:
- [tex]\( x \)[/tex] (price per gallon of midgrade gas)
- [tex]\( y \)[/tex] (price per gallon of regular gas)
From the calculations, the solutions for these equations are:
[tex]\[ x = 37.0209615384615 \][/tex]
[tex]\[ y = -121.293461538462 \][/tex]
So, the price per gallon of midgrade gas is approximately \[tex]$37.02 and the price per gallon of regular gas is approximately \$[/tex]-121.29.
### Interpretation
The results suggest that the price per gallon of regular gas is negative, which doesn't make practical sense for real-world gas prices. This might indicate a possible error in the provided data or in how it was represented. However, based on the direct solutions given:
- Price per gallon of midgrade gas: \[tex]$37.02
- Price per gallon of regular gas: \$[/tex]-121.29
It's important to review the context and the data for any potential inaccuracies or misunderstandings because normally, prices of fuel would both be positive values.
