At a farmers market, Carly buys only apples and bananas. • On Monday, she buys 4 apples and 6 bananas for a total cost of \4.25. Let be the cost of one apple, in dollars, and be the cost of one banana, in dollars. Which system of equations can be used to determine the values of and ?

Solution available with step-by-step explanation

SAT Systems of Two Linear Equations Question 22 | Free SAT Prep | Aniko

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Question 22·Easy·Systems of Two Linear Equations in Two Variables

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At a farmers market, Carly buys only apples and bananas.

• On Monday, she buys 4 apples and 6 bananas for a total cost of $3.80.
• On Friday, she buys 7 apples and 3 bananas for a total cost of $4.25.

Let $a$ be the cost of one apple, in dollars, and $b$ be the cost of one banana, in dollars.

Which system of equations can be used to determine the values of $a$ and $b$ ?

For systems-of-equations word problems like this, move step by step: first, clearly label your variables as specific quantities (here, the prices of one apple and one banana). Then, for each sentence in the prompt, create one equation by multiplying each variable by the correct number of items and setting their sum equal to the given total. Write each day’s equation separately, checking that item counts become coefficients and totals go on the other side of the equals sign. Finally, scan the choices for the system whose two equations exactly match the ones you derived, being careful not to mix up apple and banana coefficients or treat totals as coefficients.

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Focus on what $a$ and $b$ represent

$a$ and $b$ are prices, not quantities. Think: when you want the total cost for a day, how do you combine the price of one apple or banana with how many of each she buys?

Build an equation for Monday only

Use the numbers 4, 6, and 3.80. Which numbers should multiply $a$ and $b$ , and which number should be on the other side of the equals sign as the total?

Then build an equation for Friday

Repeat your thinking for 7 apples, 3 bananas, and a total of 4.25. Make sure your equation has "(something with $a$ ) + (something with $b$ ) = total cost."

Compare with the choices

Once you have one equation from Monday and one from Friday, look for the answer choice whose two equations both match what you wrote.

Desmos Guide

Represent the variables in Desmos

Decide which Desmos variables will stand for the prices: for example, let $x$ represent the cost of one apple and $y$ represent the cost of one banana. Remember this is just a stand-in for $a$ and $b$ from the problem.

Turn Monday’s sentence into an equation and enter it

From the words "4 apples and 6 bananas for a total cost of $3.80," write an equation in terms of $x$ and $y$ that adds the cost of 4 apples to the cost of 6 bananas and sets it equal to $3.80. Type that equation into Desmos (it will appear as a line).

Turn Friday’s sentence into an equation and enter it

From the words "7 apples and 3 bananas for a total cost of $4.25," write a second equation in terms of $x$ and $y$ that adds the cost of 7 apples to the cost of 3 bananas and sets it equal to $4.25. Type this equation into Desmos as well so you see two lines.

Use your equations to choose the answer

Compare the two equations you entered in Desmos with the answer choices. The correct choice is the one whose two equations match the equations you wrote and graphed. The intersection point of the two lines would give the actual prices, but for this question you only need the matching system.

Step-by-step Explanation

Identify what the variables mean

The problem tells you that:

$a$ is the cost of one apple in dollars.
$b$ is the cost of one banana in dollars.

So any equation you write should use $a$ and $b$ only as prices, never as numbers of fruits or totals.

Translate Monday’s purchase into an equation

On Monday, Carly buys 4 apples and 6 bananas and spends $3.80 total.

The cost of 4 apples is $4a$ .
The cost of 6 bananas is $6b$ .
The total cost is the sum of these two amounts, which must equal $3.80.

So the Monday information gives an equation of the form:

(cost of apples) + (cost of bananas) = total cost,
specifically, $4a + 6b = 3.80$ .

Translate Friday’s purchase into an equation

On Friday, Carly buys 7 apples and 3 bananas and spends $4.25 total.

The cost of 7 apples is $7a$ .
The cost of 3 bananas is $3b$ .
The total cost is the sum of these two amounts, which must equal $4.25.

So the Friday information gives an equation of the form:

(cost of apples) + (cost of bananas) = total cost,
specifically, $7a + 3b = 4.25$ .

Match your equations to the answer choices

Putting the two equations together, the system that represents the situation is:

4a + 6b = 3.80 \\[0.7em] 7a + 3b = 4.25

This system exactly matches choice C, so that is the correct answer.

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Step-by-step Explanation

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