Office Supply Store M sells packs of printer paper for \6.25 each. Office Supply Store N sells packs of printer paper for \5.20 each. A particular purchase of printer paper packs and ink cartridges would cost \75.50 at Store N. How many packs of printer paper are in this purchase?

Solution available with step-by-step explanation

SAT Systems of Two Linear Equations Question 63 | Free SAT Prep | Aniko

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Question 63·Hard·Systems of Two Linear Equations in Two Variables

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Office Supply Store M sells packs of printer paper for $4.75 each and ink cartridges for $6.25 each. Office Supply Store N sells packs of printer paper for $5.50 each and ink cartridges for $5.20 each. A particular purchase of printer paper packs and ink cartridges would cost $74.00 at Store M or $75.50 at Store N.

How many packs of printer paper are in this purchase?

For word problems that describe a purchase at different prices in two places, first define clear variables for each item, then write one linear equation for each store’s total cost. If there are decimals, quickly clear them by multiplying all terms by 10 or 100 to make the arithmetic cleaner. Use elimination (often by subtracting one equation from the other) to get a simple relation between the variables, solve for one variable, and substitute back to find the other. Finally, double-check that your values satisfy both original equations and match what the question is asking for (packs vs. cartridges).

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Turn the words into equations

Let $p$ be the number of paper packs and $i$ be the number of ink cartridges. Can you write one equation for the total cost at Store M and another for the total cost at Store N?

Avoid working with decimals

Once you have the two equations, multiply every price and total by 100 so you are working in cents instead of dollars. This gets rid of the decimals and makes arithmetic easier.

Combine the two equations

After you have both cent-equations, try subtracting one equation from the other to eliminate part of the expression and create a simpler equation relating $p$ and $i$ .

Finish solving the system

Use the simpler equation to express one variable in terms of the other and substitute into one of the original equations. After you find the number of cartridges, plug back in to find the number of paper packs.

Desmos Guide

Enter the two cost equations

In Desmos, let $x$ represent the number of paper packs and $y$ represent the number of ink cartridges. Type the two equations:

4.75x + 6.25y = 74
5.50x + 5.20y = 75.50

Desmos will graph these as two straight lines.

Find the intersection of the lines

Use your mouse or finger to click/tap near where the two lines cross. Desmos will display the intersection point as an ordered pair $(x,y)$ .

Interpret the intersection

Look at the $x$ -coordinate of the intersection point; this value is the number of packs of printer paper in the purchase. (The $y$ -coordinate is the number of ink cartridges.)

Step-by-step Explanation

Define variables and write the system of equations

Let $p$ be the number of packs of printer paper and $i$ be the number of ink cartridges.

At Store M, packs cost 4.75 dollars each and cartridges cost 6.25 dollars each, with a total of 74 dollars:

4.75p + 6.25i = 74

At Store N, packs cost 5.50 dollars each and cartridges cost 5.20 dollars each, with a total of 75.50 dollars:

5.50p + 5.20i = 75.50

Now we have a system of two linear equations in $p$ and $i$ .

Clear decimals and combine the equations

To make the arithmetic easier, think in cents by multiplying each equation by 100:

475p + 625i = 7400 \\[0.7em] 550p + 520i = 7550

Now subtract the first equation from the second to get a simpler relation between $p$ and $i$ :

(550p - 475p) + (520i - 625i) = 7550 - 7400 \\[0.7em] 75p - 105i = 150

Divide every term by 15 to simplify:

5p - 7i = 10

This is a simpler equation that connects $p$ and $i$ .

Solve for one variable and find the number of ink cartridges

From the simplified equation $5p - 7i = 10$ , solve for $p$ in terms of $i$ :

5p = 7i + 10 \\[0.7em] p = \frac{7i + 10}{5}

Substitute this expression for $p$ into one of the cent-equations, for example $475p + 625i = 7400$ :

475\left(\frac{7i+10}{5}\right) + 625i = 7400 \\[0.7em] 95(7i+10) + 625i = 7400 \\[0.7em] 665i + 950 + 625i = 7400 \\[0.7em] 1290i + 950 = 7400 \\[0.7em] 1290i = 6450 \\[0.7em] i = 5

So the purchase includes 5 ink cartridges. We still need to find $p$ , the number of packs of paper.

Use the simple equation to find the number of packs

Use the simpler relation $5p - 7i = 10$ with $i = 5$ :

5p - 7(5) = 10 \\[0.7em] 5p - 35 = 10 \\[0.7em] 5p = 45 \\[0.7em] p = 9

So the purchase includes 9 packs of printer paper, which corresponds to answer choice C.

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Desmos Guide

Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation