# Video: Using Linear Equations to Solve Problems

Natalie likes to do puzzles. It takes her a quarter of an hour longer to solve a hanjie than a sudoku. Over one weekend, she spent 4 1/2 hours solving puzzles. She solved 3 sudokus and 6 hanjies. Write an equation that can be used to find 𝑡, the amount of time it takes her to solve a hanjie puzzle. Solve your equation to find out how long it takes Natalie to do a hanjie puzzle.

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### Video Transcript

Natalie likes to do puzzles. It takes her a quarter of an hour longer to solve a hanjie than a sudoku. Over one weekend, she spent four and a half hours solving puzzles. She solved three sudokus and six hanjies. Write an equation that can be used to find 𝑡, the amount of time it takes her to solve a hanjie puzzle. Solve your equation to find out how long it takes Natalie to do a hanjie puzzle.

It takes Natalie 𝑡 minutes to solve one hanjie puzzle. As it takes a quarter of an hour longer to solve a hanjie than a sudoku, it would take her 𝑡 minus 15 minutes to solve one sudoku. We were told that Natalie solved three sudokus and six hanjies.

We, therefore, need to multiply 𝑡 by six and 𝑡 minus 15 by three. This gives us an expression for these nine puzzles of three multiplied by 𝑡 minus 15 plus six 𝑡. We were also told that Natalie spent a total of four and a half hours solving puzzles.

One hour is equal to 60 minutes. This means that four hours is equal to 240 minutes as 60 multiplied by four is equal to 240. Half an hour is equal to 30 minutes. We, therefore, need to add 240 and 30 minutes. Four and a half hours is equal to 270 minutes. This means that the equation we need to solve to work out time 𝑡 is three multiplied by 𝑡 minus 15 plus six 𝑡 is equal to 270.

The second part of our question asked us to solve this equation to calculate the value of 𝑡, the time taken for Natalie to do a hanjie puzzle. Expanding the bracket or parenthesis gives us three 𝑡 minus 45 as three multiplied by 𝑡 is equal to three 𝑡 and three multiplied by negative 15 is equal to negative 45. The equation is, therefore, simplified to three 𝑡 minus 45 plus six 𝑡 equals 270.

Collecting like terms gives us nine 𝑡 minus 45 equals 270 as three 𝑡 plus six 𝑡 is equal to nine 𝑡. Adding 45 to both sides of this equation gives us nine 𝑡 is equal to 315 as 270 plus 45 equals 315. Dividing both sides of this equation by nine gives us a value of 𝑡 equal to 35.

We can, therefore, say that it takes Natalie 35 minutes to complete a hanjie puzzle. As it takes a quarter of an hour longer to solve the hanjie than a sudoku, it takes Natalie 20 minutes to solve a sudoku puzzle.

We could substitute these values back into the equation to check that our answer is correct. Three multiplied by 20 plus six multiplied by 35 is equal to 270. Therefore, it does take Natalie 35 minutes to solve a hanjie puzzle and 20 minutes to solve a sudoku puzzle.