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Tom put 18 gallons of midgrade gas in his truck and filled up his empty five-gallon gas tank with regular gas for his lawnmower at home. He spent [tex]$\$[/tex]59.91[tex]$.

The following week, he put 14 gallons of midgrade gas in his truck and topped off his five-gallon tank with just one gallon of regular gas. If he paid $[/tex]\[tex]$39.70$[/tex] and the prices remained the same, find the price per gallon of midgrade and regular gas.


Sagot :

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.