1. What is the price of a danish and a filled donut? 2. How many solutions does the system of equations have? 3. Solve the system of equations using the substitution method. 4. Wri... 1. What is the price of a danish and a filled donut? 2. How many solutions does the system of equations have? 3. Solve the system of equations using the substitution method. 4. Write an equation that models total points scored in a football game. 5. What is the combined cost of one notebook and one pen? Read more

Steps to Solve

1. Find the price of a danish To find the price of a danish, we can set up an equation based on the information given. Let ( d ) represent the price of a danish and ( f ) the price of a filled donut.

Bill bought 4 danishes and 9 filled donuts for $14.20, leading to the equation: $$ 4d + 9f = 14.20 $$

Mary Ann bought 6 danishes for $12.85: $$ 6d = 12.85 $$

Now, solve for ( d ): $$ d = \frac{12.85}{6} = 2.14167 \text{ (approximately $2.14)} $$

2. Substitute to find the price of a filled donut Now that we have the price of a danish: Substituting ( d = 2.14 ) into the first equation to solve for ( f ): $$ 4(2.14) + 9f = 14.20 $$ Calculating ( 4(2.14) ): $$ 8.56 + 9f = 14.20 $$

3. Isolate and solve for ( f ) Subtract ( 8.56 ) from both sides: $$ 9f = 14.20 - 8.56 $$ $$ 9f = 5.64 $$ Now, divide by 9: $$ f = \frac{5.64}{9} = 0.62667 \text{ (approximately $0.63)} $$

4. Summary of prices The price of a danish is approximately $2.14 and the price of a filled donut is approximately $0.63.

5. Determine the solutions for the equations For the system of equations, we need to analyze: $$ y = -\frac{5}{2}x + 4 $$ $$ 5x + 2y = 11 $$

Rearranging the second equation for substitution: $$ 2y = 11 - 5x $$ $$ y = \frac{11 - 5x}{2} $$

Set this equal to the first equation: $$ -\frac{5}{2}x + 4 = \frac{11 - 5x}{2} $$

Multiply by 2 to eliminate the fraction: $$ -5x + 8 = 11 - 5x $$

Since both sides are equal, this system has infinitely many solutions.

6. Write the equation modeling points in the football game Given the scoring system: Let ( x ) be the number of field goals and ( y ) be the number of touchdowns. The total points can be modeled with: $$ 3x + 7y = 34 $$

7. Combined cost of one notebook and one pen Let the cost of one pen be ( p ). Then the cost of one notebook is ( p + 1.75 ): The equation becomes: $$ 6(p + 1.75) + 3p = 35 $$

Expanding and solving: $$ 6p + 10.50 + 3p = 35 $$ $$ 9p + 10.50 = 35 $$ $$ 9p = 24.50 $$ $$ p = \frac{24.50}{9} \approx 2.72 $$

The cost of one notebook is approximately ( 2.72 + 1.75 = 4.47 ).