Math Problem Statement

The system of linear equations describes a dressmaker who buys a total of 4.25 yards of silk and cotton fabric, spending $36.35. Silk costs $16.90 per yard, and cotton costs $4 per yard. The system of equations is: x + y = 4.25 and 16.90x + 4y = 36.35.

Solution

The image contains a system of linear equations related to a fabric sale. The context involves purchasing silk and cotton fabric, where silk costs $16.90 per yard and cotton costs $4 per yard. A dressmaker spent a total of $36.35 on 4.25 yards of fabric combined.

The system of equations given is:

x + y &= 4.25 \\ 16.90x + 4y &= 36.35 \end{align*}$$ Where: - $$x$$ represents the number of yards of silk purchased. - $$y$$ represents the number of yards of cotton purchased. ### 1. **What does the solution to the system represent?** The solution to this system represents the exact quantities (in yards) of silk $$x$$ and cotton $$y$$ that the dressmaker purchased to satisfy the given constraints: buying 4.25 total yards of fabric and spending $36.35. ### 2. **Find the solution to the system of equations.** We can solve this system using substitution or elimination. Let’s solve this step-by-step using substitution: 1. From the first equation $$x + y = 4.25$$, solve for $$y$$: $$y = 4.25 - x$$ 2. Substitute $$y = 4.25 - x$$ into the second equation: $$16.90x + 4(4.25 - x) = 36.35$$ Simplifying: $$16.90x + 17 - 4x = 36.35$$ Combine like terms: $$12.90x + 17 = 36.35$$ Subtract 17 from both sides: $$12.90x = 19.35$$ Divide by 12.90: $$x \approx 1.5$$ 3. Substitute $$x = 1.5$$ back into $$y = 4.25 - x$$: $$y = 4.25 - 1.5 = 2.75$$ ### Final Answer: The dressmaker bought 1.5 yards of silk and 2.75 yards of cotton. Would you like further details or explanations? Here are five related questions: 1. How would you solve the system using the elimination method? 2. What is the graphical interpretation of this system of equations? 3. How can you check if your solution is correct by plugging it into both equations? 4. How would the solution change if the price of cotton was different? 5. Can you create a similar problem involving a different set of constraints? **Tip:** For systems of linear equations, substitution works well when one equation is already solved for a variable, while elimination is efficient when the variables align for easy cancellation.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Systems of Equations

Formulas

x + y = 4.25
16.90x + 4y = 36.35

Theorems

Substitution Method
Elimination Method

Suitable Grade Level

Grades 8-10

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