Comparing Two Plans with a System of Equations | Grade 8 Math | LevelKite

Many prices have two parts: a fixed fee that stays the same no matter how much you use, and a rate, which is the price for each unit you use, such as each visit or each hour. A gym might charge a monthly fee plus a price for every visit. In this lesson, a plan is a pricing rule like that.

When you choose between two plans, asking which one is cheaper is not enough. The better question is: cheaper for how much use? One plan can cost less when you use a little, while the other costs less when you use a lot. You already know how to turn a word problem into two equations and solve them together. Now you will use that skill to find where two plans cost the same, and to decide which plan costs less on each side of that amount.

Here are two plans at a climbing gym, each for one month.

Member plan: a fee of 30 dollars, plus 4 dollars for each visit.
Pay-as-you-go plan: no fee, and 10 dollars for each visit.

Before you write an equation, say what the letters mean. Let $x$ be the number of visits in the month, and let $y$ be the total cost for the month in dollars. The domain is the set of values that $x$ is allowed to take, and those values are called permitted values. Here $x$ must be a whole number that is $0$ or greater, because you cannot make a negative number of visits or part of a visit.

Each plan gives one equation: total cost equals the fee plus the rate times the number of visits. The cost rises by the same amount for every added visit, so each one is a linear equation. Two equations that use the same variables and must be true together form a system of equations.

\begin{cases} y = 30 + 4x & \text{Member plan} \\ y = 10x & \text{Pay-as-you-go plan} \end{cases}

Total cost for one month, in dollars

Number of visits	Member plan	Pay-as-you-go plan	Plan that costs less
$0$	$30 + 4(0) = 30$	$10(0) = 0$	Pay-as-you-go
$2$	$30 + 4(2) = 38$	$10(2) = 20$	Pay-as-you-go
$4$	$30 + 4(4) = 46$	$10(4) = 40$	Pay-as-you-go
$6$	$30 + 4(6) = 54$	$10(6) = 60$	Member
$8$	$30 + 4(8) = 62$	$10(8) = 80$	Member

Look down the last column. At 4 visits the Pay-as-you-go plan costs less, and at 6 visits the Member plan costs less. The cheaper plan switches somewhere between 4 and 6 visits, but the table skips amounts, so it cannot show exactly where.

The switch is marked by the value of $x$ where both equations give the same $y$ . A pair $(x, y)$ that makes both equations true is a solution of the system, so solving the system finds the switch exactly instead of by guessing.

There is only one such value here. Each added visit adds 4 dollars to the Member plan and 10 dollars to the Pay-as-you-go plan, so the gap between the two costs changes by 6 dollars with every visit. A gap that keeps changing by the same nonzero amount can be zero at only one value of $x$ .

Key concept

The break-even point separates two plans

Write each plan as an equation, total cost $y$ equals fee plus rate times amount $x$ , using the same $x$ and $y$ for both plans. To find where the costs match, set the two costs equal and solve. When the two rates are different, the system has exactly one solution $(x, y)$ , called the break-even point. If its $x$ is a permitted value, both plans cost $y$ at that amount.

The break-even $x$ separates the permitted amounts into two sides. For amounts less than it, the plan with the greater rate costs less. For amounts greater than it, the plan with the smaller rate costs less. Test a permitted amount from a side to confirm which plan costs less there.

\text{fee}_A + \text{rate}_A \cdot x = \text{fee}_B + \text{rate}_B \cdot x

Worked example

Example 1: The climbing gym plans

The Member plan is $y = 30 + 4x$ and the Pay-as-you-go plan is $y = 10x$ . Here $x$ is the number of visits in a month, a whole number that is $0$ or greater, and $y$ is the total cost in dollars. Find the break-even point. Then say which plan costs less for each number of visits.

At the break-even point, both equations give the same y for the same x, so the two expressions for y are equal there. Set them equal to get one equation that has only x in it.
$30 + 4x = 10x$
Subtract 4x from both sides so that x is on one side only. Subtracting the same amount from both sides keeps the same solutions. On the right, subtracting 4x from 10x leaves 6x, an equivalent expression: it has the same value for every x.
$\begin{aligned} 30 + 4x - 4x &= 10x - 4x \\ 30 &= 6x \end{aligned}$
Divide both sides by 6 to get x alone. Dividing both sides by the same number keeps the same solutions as long as that number is not zero, and 6 is not zero.
$\begin{aligned} \frac{30}{6} &= \frac{6x}{6} \\ 5 &= x \end{aligned}$
Find y by putting x = 5 into the Pay-as-you-go equation. Then check the pair in the Member equation as well, because a solution must make both original equations true. Both give 50. Also, 5 is a whole number, so it is a permitted value.
$\begin{aligned} \text{Pay-as-you-go: } y &= 10(5) = 50 \\ \text{Member: } y &= 30 + 4(5) = 30 + 20 = 50 \end{aligned}$
Test one permitted amount on each side of 5. At 4 visits the Pay-as-you-go plan costs less, 40 dollars against 46. At 6 visits the Member plan costs less, 54 dollars against 60. The costs are equal only at 5, so the cheaper plan cannot change again on either side.
$\begin{aligned} x = 4: &\quad 30 + 4(4) = 46 \text{ and } 10(4) = 40 \\ x = 6: &\quad 30 + 4(6) = 54 \text{ and } 10(6) = 60 \end{aligned}$

Answer

The break-even point is $(5, 50)$ : at 5 visits both plans cost 50 dollars. For fewer than 5 visits the Pay-as-you-go plan costs less, and for more than 5 visits the Member plan costs less.

Worked example

Example 2: Two bike rental shops

Two shops rent bikes and charge for the exact time you ride, including parts of an hour. Shop A charges a fee of 5 dollars plus 3 dollars per hour. Shop B charges a fee of 11 dollars plus 1.50 dollars per hour. For what ride time do the two shops cost the same? Which shop costs less for a ride of 2.5 hours?

Name the variables and the domain. Let x be the ride time in hours, where x can be any number that is 0 or greater. Let y be the total cost in dollars. Write one equation for each shop: the fee plus the rate times the hours.
$\begin{cases} y = 5 + 3x & \text{Shop A} \\ y = 11 + 1.5x & \text{Shop B} \end{cases}$
Both expressions equal y at the break-even point, so set them equal to each other.
$5 + 3x = 11 + 1.5x$
Subtract 1.5x from both sides, then subtract 5 from both sides. Each time, you take the same amount from both sides, so the solutions stay the same.
$\begin{aligned} 5 + 3x - 1.5x &= 11 + 1.5x - 1.5x \\ 5 + 1.5x &= 11 \\ 5 + 1.5x - 5 &= 11 - 5 \\ 1.5x &= 6 \end{aligned}$
Divide both sides by 1.5. This keeps the same solutions because 1.5 is not zero. To check the division, multiply back: 1.5 times 4 is 6.
$x = \frac{6}{1.5} = 4$
Find y from Shop A's equation, then check the pair in Shop B's equation. Both original equations give 17. Also, 4 hours is a permitted time, so the shops really do cost the same there.
$\begin{aligned} \text{Shop A: } y &= 5 + 3(4) = 5 + 12 = 17 \\ \text{Shop B: } y &= 11 + 1.5(4) = 11 + 6 = 17 \end{aligned}$
A ride of 2.5 hours is permitted, and 2.5 is less than the break-even time of 4. Find both costs at x = 2.5 and compare them. Shop A has the greater rate, and it costs less on this side, as the key idea says.
$\begin{aligned} \text{Shop A: } 5 + 3(2.5) &= 5 + 7.5 = 12.5 \\ \text{Shop B: } 11 + 1.5(2.5) &= 11 + 3.75 = 14.75 \end{aligned}$

Answer

The two shops cost the same, 17 dollars, for a ride of exactly 4 hours. For a ride of 2.5 hours, Shop A costs 12.50 dollars and Shop B costs 14.75 dollars, so Shop A costs less by 2.25 dollars.

Worked example

Example 3: A break-even amount that is not permitted

A pool sells two summer passes. Pass A costs 45 dollars plus 2 dollars for each swim. Pass B costs 15 dollars plus 6 dollars for each swim. Let $x$ be the number of swims in the summer, a whole number that is $0$ or greater, and let $y$ be the total cost in dollars. Which pass costs less for each number of swims?

Write one equation for each pass. Both expressions equal y at the break-even point, so set them equal.
$\begin{aligned} y &= 45 + 2x \quad \text{(Pass A)} \\ y &= 15 + 6x \quad \text{(Pass B)} \\ 45 + 2x &= 15 + 6x \end{aligned}$
Subtract 2x from both sides, then subtract 15 from both sides. The same amount comes off both sides each time, so the solutions stay the same.
$\begin{aligned} 45 + 2x - 2x &= 15 + 6x - 2x \\ 45 &= 15 + 4x \\ 45 - 15 &= 15 + 4x - 15 \\ 30 &= 4x \end{aligned}$
Divide both sides by 4, which is allowed because 4 is not zero. The result, 7.5, is exact, not rounded.
$x = \frac{30}{4} = 7.5$
Find y from Pass A's equation and check the pair in Pass B's equation. Both give 60, so (7.5, 60) solves the system. But x must be a whole number, and 7.5 is not a whole number. So no permitted number of swims makes the two passes cost the same.
$\begin{aligned} \text{Pass A: } y &= 45 + 2(7.5) = 45 + 15 = 60 \\ \text{Pass B: } y &= 15 + 6(7.5) = 15 + 45 = 60 \end{aligned}$
The value 7.5 still marks where the cheaper pass switches. Test the nearest permitted amount on each side, 7 swims and 8 swims. At 7 swims Pass B costs less, 57 dollars against 59. At 8 swims Pass A costs less, 61 dollars against 63. The rates are different, so the cheaper pass stays the same across each whole side.
$\begin{aligned} x = 7: &\quad 45 + 2(7) = 59 \text{ and } 15 + 6(7) = 57 \\ x = 8: &\quad 45 + 2(8) = 61 \text{ and } 15 + 6(8) = 63 \end{aligned}$

Answer

The passes never cost exactly the same, because the break-even amount, 7.5 swims, is not a whole number. For 7 swims or fewer, Pass B costs less. For 8 swims or more, Pass A costs less.

In every example so far, the two rates were different, and the key idea needs that. Here is why: when you collect the $x$ terms and solve, your last step divides by the difference of the rates, and you may never divide by zero.

Suppose two plans have the same rate: $y = 20 + 5x$ and $y = 35 + 5x$ . Set the costs equal to get $20 + 5x = 35 + 5x$ , then subtract $5x$ from both sides to get $20 = 35$ . That statement is false for every $x$ . An equation that no value makes true is called a contradiction. No permitted amount makes these plans cost the same, and the first plan costs 15 dollars less for every permitted $x$ .

Now suppose two plans have the same rate and the same fee. The same step gives a statement such as $20 = 20$ , which is true for every $x$ . An equation that every permitted value makes true is called an identity. These plans cost the same for every permitted amount.

To compare two plans, write one equation for each plan with the same variables, state the domain, and solve the system. When the rates are different, the system has exactly one solution, the break-even point. It marks where the cheaper plan switches, and the plans cost the same there only if that amount is permitted. Check the solution in both original equations, then test a permitted amount on a side to see which plan costs less on that whole side. Next, you will use systems of equations to solve rate and mixture problems.