Kate and Nate play a game in which they earn the same number of points for each goal and lose the same number of points for each penalty. Kate makes 6 goals and 3 penalties, ending the game with 6 points. Nate earns 8 goals and 9 penalties and ends the game with -22. Write a system of equations that describe Kate and Nate's outcomes. Use x to represent the number of points for a goal and y to represent the number of points for a penalty.

Solve the problem by using a venn diagram a group of students were surveyed about their taste in music. they were asked how they like three types of music: rock, country western and jazz. the results are summarized below: no one dislikes all three types of music. six like all three types of music. eleven don’t like country western. sixteen like jazz. three like country western and jazz, but not rock. six don’t like rock. eight like rock and country western. seven don’t like jazz. how many students were surveyed?