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Equivalent Equations

Sometimes the secret to an algebra problem or word problem is setting two equations equal to each other. 

Sometimes the problem will give you the equations. The classic example of this is when a problem gives you an equation for revenue and an equation for expenses, and asks for a break even point.

Example:

A pillow company makes luxurious down pillows. The cost of a run of pillows can be represented by the function $c(x)=340+12(x)+3(x)$, where $x$ equals the number of pillows made in a run. The revenue from the pillows can be represented with the function $r(x)=56(x)-120$.  How many pillows does the company need to make to break even?

"Break even" means expenses and revenues are equal. So, set the expense equation and the revenue equations equal and solve for $x$:

$\eqalign{340+12x+3x&=56x-120\\340+15x&=56x-120\\+120\qquad&=\quad -120\\220+15x&=56x\\-15x&=-15x\\\dfrac{220}{41}&=\dfrac{41x}{41}\\x&=\dfrac{220}{41}=5.366}$

They need to sell 5.366 pillows.

 

Other times, you have to write the equations before you set the equations equal.  See Create Equations (Unknown in Terms of Same Variable) for a review on setting up the equations.

Example:

Jason has 60 in the bank. He gets 4 dollars a week allowance. His younger brother, Abel, has 100 dollars in the back but only gets 2 dollars a week in allowance. If neither of them spends any money, in how many weeks will hey have the same amount of money?

Jason=$60+4x$

Abel=$100+2x$

When will they be equal?

$\eqalign{60+4w&=100+2x\\-2w&=\quad -2w\\60+2w=100\\-60 \quad&=-60\\2w&=40\\w&=20}$

They will have the same amount of money in 20 weeks.

 

These equations sometimes get complicted. But, once you get them written, always think about whether setting them equal will help you get to the final solution.

Practice Problems:

  • Equivalent Equations

    $P(c)=60c-35c-26$

    $P(b)=1.25b-10$

    1. A company sells both handmade stocking caps and mass produced beanies. The functions above show how much profit $P(c)$ they make on $c$ handmade stocking caps and how much profit $P(b)$ they make on $b$ beanies. The beanies are much cheaper to buy, but they also yield a much smaller profit. How many beanies do they have to sell to equal the profit from 25 handmade stockingcaps.

     

    $C(m)=45+2.70(\dfrac{m}{20})$

    $A(m)=300+.05m$

    2. The functions above show the cost of a trip by car $C(m)$ and by airplane $A(m)$ based upon the miles $m$ of the trip. For a trip for which it would cost the same amount to fly or drive, how much would it cost to fly?

     

    $A(x)=150+.55x$

    $P(x)=230+.35x$

    3. The functions above the cost of growing $x$ number of apples and $x$ number of pears. For the quantity of fruit for which it costs the same to grow apples or pears, how much does it cost to grow apples?

     

    4. At a cafe, coffee costs \$1.25 a cup, and lattes cost twice as much as coffee. Donut holes are 35 cents each, and donuts are 90 cents. If a mom orders a latte and a donut, and tells her child he can spend the same amount of money (but no more), how many donut holes can he buy?

     

    5.  A YMCA preschool charges a \$100 enrollment fee and \$795 a month in tuition. A local church preschool charges no enrollment fee but charges \$800 a month plus a 3% surcharge on that tuition each month. After how many months is it more cost efficient to attend the YMCA rather than the church preschool?

Common Core Grade Level/Subject