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Math Problem: Limes & Pears Linear Equations

Jan 27, 2026 by ADMIN 45 views

Math Problem: Limes & Pears Linear Equations

Hey guys, let's dive into a cool math problem that involves a bit of shopping and a whole lot of algebra! We're going to break down how to set up a system of linear equations to model a real-world scenario. So, imagine Mariah goes to the store and she's got a budget of $9.50. She wants to buy two kinds of fruit: limes and pears. She buys a total of 9 pounds of these fruits. Now, here's the kicker: limes are super cheap at $0.50 per pound, while pears are a bit pricier at $1.50 per pound. Our mission, should we choose to accept it, is to figure out the system of linear equations that accurately represents this situation. We'll use '$ l $' to stand for the number of pounds of limes and '$ p $' to stand for the number of pounds of pears. This problem is a classic example of how math can help us organize and solve everyday dilemmas, like figuring out exactly how much of each fruit Mariah could buy without breaking the bank or going over her desired weight. We need to create two distinct equations that capture both the total weight and the total cost of her purchase. It's like creating a mathematical blueprint for her shopping trip! We'll be using the variables '$ l $' and '$ p

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to represent our unknowns, which is standard practice in algebra. The goal is to have a set of equations that, when solved, will tell us precisely how many pounds of limes and how many pounds of pears Mariah bought. Stick around, and we'll walk through it step-by-step, making sure you understand the logic behind each equation.

Understanding the Variables and the Scenario

First off, let's get super clear on what our variables represent. We've got ' $l$ ' for the pounds of limes and ' $p$ ' for the pounds of pears. These are the two quantities we don't know yet, and our system of equations will help us find them. The problem states that Mariah bought a total of 9 pounds of fruit. This is a key piece of information, and it directly relates to the quantities of limes and pears she purchased. So, if you add the weight of the limes ( ' $l$ ' pounds) and the weight of the pears (' $p$ ' pounds), the sum must equal 9 pounds. This gives us our first equation. Think of it as the 'total weight' equation. It's pretty straightforward: the amount of limes plus the amount of pears equals the total amount of fruit. This equation will be fundamental in solving our problem because it provides one constraint on the possible values of ' $l$ ' and ' $p$ '. We're dealing with a linear equation here because both variables, ' $l$ ' and ' $p$ ', are raised to the power of 1, and there are no products of variables (like ' $l imes p$ '). This keeps things nice and simple, as linear equations represent straight lines when graphed, and they are generally easier to solve than more complex equations. It’s crucial to identify these direct relationships between the variables and the given totals in word problems. This first equation is a direct translation of the statement about the total weight of the produce Mariah bought. Without this, we'd have way too many possibilities for the amounts of limes and pears.

Crafting the Total Weight Equation

So, building on what we just discussed, let's formalize the first equation. We know that the number of pounds of limes, represented by ' $l$ ', added to the number of pounds of pears, represented by ' $p$ ', must equal the total weight of the fruit purchased, which is 9 pounds. Therefore, our first linear equation is: $l + p = 9$ . This equation is super important because it establishes a direct relationship between the quantities of the two fruits. It tells us that if Mariah buys, say, 3 pounds of limes, she must buy 6 pounds of pears to reach the total of 9 pounds (because 3 + 6 = 9). Or, if she buys 5 pounds of pears, she must buy 4 pounds of limes (because 4 + 5 = 9). This equation is the foundation of our system. It's a linear equation because both ' $l$ ' and ' $p$ ' are first-degree variables (they don't have exponents higher than 1), and there are no products of variables. In graphical terms, this equation would represent a straight line on a coordinate plane where the x-axis is ' $l$ ' and the y-axis is ' $p$ ' (or vice-versa). This is why we call it a linear equation – it describes a linear relationship. Understanding this equation is the first big step in modeling Mariah's shopping trip accurately. It captures the physical constraint of the total weight of the fruit she bought. Without this equation, we couldn't possibly solve for the exact amounts of each fruit. It's the bedrock upon which we build our mathematical model.

Incorporating the Cost: The Second Equation

Now, let's tackle the money aspect, which will give us our second linear equation. Mariah spent a total of $9.50. This total cost is made up of the cost of the limes plus the cost of the pears. We know the price per pound for each fruit: limes are $0.50 per pound, and pears are $1.50 per pound. So, to find the total cost of the limes, we multiply the number of pounds of limes ('$ l

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) by the cost per pound of limes ($0.50). This gives us

0.50l

. Similarly, to find the total cost of the pears, we multiply the number of pounds of pears ('

p

') by the cost per pound of pears ($1.50). This gives us

1.50p

. Since the total amount of money spent is $9.50, the sum of the cost of the limes and the cost of the pears must equal $9.50. This gives us our second linear equation: $0.50l + 1.50p = 9.50$ . This equation is also linear for the same reasons as the first one: '

l

' and '

p

' are first-degree variables, and there are no products of variables. It represents the financial constraint of Mariah's purchase. It tells us about the value of the items she bought. Together with the first equation (

l + p = 9

), this second equation creates a system that precisely models the situation. Each equation represents a different aspect of Mariah's purchase – one about the quantity and the other about the cost. This is the power of systems of equations; they allow us to model situations with multiple conditions simultaneously. We're getting closer to solving the whole puzzle, guys!

The Complete System of Linear Equations

Alright, we've done the heavy lifting! We've successfully translated the word problem into two distinct linear equations. The first equation, derived from the total weight of the fruit, is $l + p = 9$ . This equation represents the constraint that the sum of the pounds of limes (' $l$ ') and the pounds of pears (' $p$ ') must equal 9 pounds. It's all about the quantity. The second equation, derived from the total cost of the fruit, is $0.50l + 1.50p = 9.50$ . This equation represents the financial constraint, stating that the cost of the limes ( ' $0.50l$ ' ) plus the cost of the pears ( ' $1.50p$ ' ) must equal the total amount spent, which is $9.50. It's all about the value. When we put these two equations together, we get what's called a system of linear equations:

l + p = 9
0.50l + 1.50p = 9.50

This is the mathematical model that perfectly describes Mariah's shopping scenario. Each equation is a line, and the solution to the system (if one exists and is unique) would be the point where these two lines intersect on a graph. This intersection point represents the specific values of ' $l$ ' and ' $p$ ' that satisfy both conditions simultaneously – meaning, the exact pounds of limes and pears that add up to 9 pounds and cost exactly $9.50. This system is essential for determining the exact quantities of each fruit Mariah purchased. It's the answer to the question posed in the problem: Which system of linear equations models the situation? This is a fantastic example of how algebra helps us break down complex real-world problems into manageable mathematical components. Pretty neat, right?

Solving the System (Optional but Helpful!)

While the question only asks for the system of equations, it's often helpful to see how to solve it to truly appreciate the model. We have our system:

$l + p = 9$
$0.50l + 1.50p = 9.50$

We can use substitution or elimination to solve this. Let's use substitution. From equation (1), we can easily isolate ' $l$ ': $l = 9 - p$ . Now, we substitute this expression for ' $l$ ' into equation (2):

$0.50(9 - p) + 1.50p = 9.50$

Distribute the 0.50:

$4.50 - 0.50p + 1.50p = 9.50$

Combine the ' $p$ ' terms:

$4.50 + 1.00p = 9.50$

Subtract 4.50 from both sides:

$1.00p = 5.00$

So, $p = 5$ pounds of pears.

Now, substitute this value of ' $p$ ' back into our equation for ' $l$ ' ( $l = 9 - p$ ):

$l = 9 - 5$

$l = 4$ pounds of limes.

So, Mariah bought 4 pounds of limes and 5 pounds of pears. Let's check: Total pounds = 4 + 5 = 9 pounds (correct!). Total cost = ( $0.50 imes 4$ ) + ( $1.50 imes 5$ ) = $2.00 + $7.50 = $9.50 (correct!). This confirms that our system of equations accurately modeled the scenario and that our solution is correct. Seeing the actual amounts makes the math feel much more concrete, doesn't it?

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