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According to the question, there are two rental plans.

The cost of renting a car as per Standard Rental plan is $52 per day plus 23 cents per mile. In the Deluxe Rental Plan, the cost to rent a car is $80 per day with unlimited mileage.

Consider two variables x and y.

Denote the cost of renting a car by y and number of miles driven by x.

Let the number of miles driven be x.

In Standard Rental plan car is rented for$52 per day plus 23 cents per mile. Therefore, the cost to rent a car is expressed below. The per day cost is fixed as one-day trip is to be scheduled.

$y=0.23x+52$

In Deluxe Rental plan car is rented for$80 per day with unlimited mileage.

Therefore, the cost to rent a car is expressed below.

$y=80$

Thus, the equation that represent cost of renting a car as per Standard Rental plan is $y=0.23x+52$ and cost of renting a car as per Deluxe Rental plan is $y=80$.

Equation of line in slope intercept form is expressed below.

$y=mx+c$

Where m is the slope and c is the intercept of y-axis.

Consider the first equation $y=0.23x+52$.

Now, the equation is in the form $y=mx+c$. Here slope m of the line is $0.23$ and intercept of y-axis c is $52$.

Now, consider the second equation $y=80$.

Rewrite the equation in form of slope-intercept form.

$y=0x+80$

Now, the equation is in the form $y=mx+c$. Here slope m of the line is 0 and intercept of y-axis c is 80.

Plot the equations on the same plane and the point where both the equations intersect is the break-even point of the system of the equations.

The red line denotes the equation $y=0.23x+52$ and blue line denotes the equation $y=80$.

Therefore, the break-even point of the rental costs is $\left(121.7,80\right)$.