Beatrice bought 3 metres of denim fabric and 5 metres of cotton fabric. The total bill, excluding tax, was $22. Candace bought 2 metres of cotton and 6 metres of denim for $28.
a) Write a linear system to find the price of the cotton and denim fabric.
b) Solve the system by graphing: [tex]3 x+5 y=22[/tex]
c) How much will 8 metres of Denim & 5 metres of cotton cost?
Answer (2)
Define variables: Let x be the price per metre of denim and y be the price per metre of cotton.
Formulate equations: 3 x + 5 y = 22 and 6 x + 2 y = 28 .
Solve the system: Using elimination, find x = 4 and y = 2 .
Calculate the total cost: 8 ( 4 ) + 5 ( 2 ) = 42 . The cost is 42 .
Explanation
Understanding the Problem Let's break down this problem step by step so we can understand how to solve it! We're given information about two purchases of denim and cotton fabric, and we need to find the price per metre of each fabric. Then, we'll use those prices to calculate the cost of a new purchase.
Defining Variables Let's define our variables:
Let x be the price per metre of denim fabric. Let y be the price per metre of cotton fabric.
Writing the Linear System Now, we can write a system of linear equations based on the information given:
Beatrice's purchase: 3 x + 5 y = 22 Candace's purchase: 6 x + 2 y = 28
Solving for y To solve this system, we can use the substitution or elimination method. Let's use the elimination method. We can multiply the first equation by -2 to eliminate x :
− 2 ( 3 x + 5 y ) = − 2 ( 22 ) − 6 x − 10 y = − 44
Now we have two equations:
− 6 x − 10 y = − 44 6 x + 2 y = 28
Adding these two equations, we get:
− 8 y = − 16 y = 2
Solving for x Now that we have the value of y , we can substitute it back into one of the original equations to solve for x . Let's use the first equation:
3 x + 5 ( 2 ) = 22 3 x + 10 = 22 3 x = 12 x = 4
Finding the Prices So, the price of denim fabric is $4 per metre, and the price of cotton fabric is $2 per metre.
Calculating the Total Cost Finally, we can calculate the cost of 8 metres of denim and 5 metres of cotton:
Cost = 8 x + 5 y Cost = 8 ( 4 ) + 5 ( 2 ) Cost = 32 + 10 Cost = 42
Final Answer Therefore, 8 metres of denim and 5 metres of cotton will cost $42.
Examples
Understanding systems of equations is incredibly useful in everyday life. For example, imagine you're planning a party and need to buy snacks. You know the total cost and the quantities of two different snacks you purchased. By setting up a system of equations, you can determine the individual price of each snack. This same principle applies to budgeting, comparing costs of different services, or even optimizing your workout routine by balancing different types of exercises.
To determine the cost of denim and cotton fabric, we set up a system of linear equations based on two purchases. After solving the equations, we find that the cost of 8 metres of denim and 5 metres of cotton fabric totals $42. Graphing the equations can visually show where they intersect, confirming our calculated prices.
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