Question

1. at a local store, 4 bagels and 4 cups of coffee cost $9.20. the cost of 5 bagels and 6 cups of coffee is $12.80. set up and solve a system of equations to find the cost of a bagel and the cost of a cup of coffee.. use desmos.com to graph and solve. let statements: equations: solution: interpret:

Explanation:

Let Statements:

Let \( x \) be the cost of a bagel (in dollars) and \( y \) be the cost of a cup of coffee (in dollars).

Equations:

From "4 bagels and 4 cups of coffee cost $9.20", we get:
\( 4x + 4y = 9.20 \)

From "5 bagels and 6 cups of coffee cost $12.80", we get:
\( 5x + 6y = 12.80 \)

Solution:

Step 1: Simplify the first equation

Divide \( 4x + 4y = 9.20 \) by 4:
\( x + y = 2.30 \)
Rearrange to solve for \( x \):
\( x = 2.30 - y \)

Step 2: Substitute \( x \) into the second equation

Substitute \( x = 2.30 - y \) into \( 5x + 6y = 12.80 \):
\( 5(2.30 - y) + 6y = 12.80 \)

Step 3: Expand and simplify

Expand: \( 11.50 - 5y + 6y = 12.80 \)
Combine like terms: \( 11.50 + y = 12.80 \)

Step 4: Solve for \( y \)

Subtract 11.50 from both sides:
\( y = 12.80 - 11.50 = 1.30 \)

Step 5: Solve for \( x \)

Substitute \( y = 1.30 \) into \( x = 2.30 - y \):
\( x = 2.30 - 1.30 = 1.00 \)

Interpret:

The cost of one bagel is \( \boldsymbol{\$1.00} \), and the cost of one cup of coffee is \( \boldsymbol{\$1.30} \).

(To verify with Desmos: Graph \( y = -\frac{4}{4}x + \frac{9.20}{4} \) (simplified to \( y = -x + 2.3 \)) and \( y = -\frac{5}{6}x + \frac{12.80}{6} \) (or \( y = -\frac{5}{6}x + \frac{64}{30} \approx -\frac{5}{6}x + 2.133 \)). The intersection point is \( (1.00, 1.30) \), confirming the solution.)

Answer:

Let Statements:

Let \( x \) be the cost of a bagel (in dollars) and \( y \) be the cost of a cup of coffee (in dollars).

Equations:

From "4 bagels and 4 cups of coffee cost $9.20", we get:
\( 4x + 4y = 9.20 \)

From "5 bagels and 6 cups of coffee cost $12.80", we get:
\( 5x + 6y = 12.80 \)

Solution:

Step 1: Simplify the first equation

Divide \( 4x + 4y = 9.20 \) by 4:
\( x + y = 2.30 \)
Rearrange to solve for \( x \):
\( x = 2.30 - y \)

Step 2: Substitute \( x \) into the second equation

Substitute \( x = 2.30 - y \) into \( 5x + 6y = 12.80 \):
\( 5(2.30 - y) + 6y = 12.80 \)

Step 3: Expand and simplify

Expand: \( 11.50 - 5y + 6y = 12.80 \)
Combine like terms: \( 11.50 + y = 12.80 \)

Step 4: Solve for \( y \)

Subtract 11.50 from both sides:
\( y = 12.80 - 11.50 = 1.30 \)

Step 5: Solve for \( x \)

Substitute \( y = 1.30 \) into \( x = 2.30 - y \):
\( x = 2.30 - 1.30 = 1.00 \)

Interpret:

The cost of one bagel is \( \boldsymbol{\$1.00} \), and the cost of one cup of coffee is \( \boldsymbol{\$1.30} \).

(To verify with Desmos: Graph \( y = -\frac{4}{4}x + \frac{9.20}{4} \) (simplified to \( y = -x + 2.3 \)) and \( y = -\frac{5}{6}x + \frac{12.80}{6} \) (or \( y = -\frac{5}{6}x + \frac{64}{30} \approx -\frac{5}{6}x + 2.133 \)). The intersection point is \( (1.00, 1.30) \), confirming the solution.)