In a certain game, a player can solve easy or hard puzzles. A player earns 30 points for solving an easy puzzle and 60 points for solving a hard puzzle. Tina solved a total of 50 puzzles playing this game, earning 1,950 points in all. How many hard puzzles did Tina solve? A) 10 B) 15 C) 25 D) 35

This is a system of equations. We can solve it by recognizing the two linear equalities we've given to solve this equation.First, let's start off by recognizing two variables: x and y. x will be our number of hard puzzles, and y will be our number of easy puzzles. We're given that the number of hard puzzles and easy puzzles sum to 50, so we can rewrite that as:[tex]x+y=50[/tex]Next, we're given that the sum of the points gained from the hard puzzles (60x), and the number of points gained from the easy puzzles (30y), sum to 1950. We can rewrite that as:[tex]60x+30y=1950[/tex]Now, we have our two linear equations, and we must solve for the hard puzzles, so x. Solving for the hard puzzles means eliminating the easy puzzles from our systems, so we can multiply our first equation by 30, and subtract it from the first equation. This is a technique called elimination.[tex]30x+30y=1500 \\ 60x+30y=1950 \\ 30x=450 \implies x=15 [/tex]Since x=15, Tina has solved 15 hard puzzles.