Linear equations in two variables are fundamental tools in mathematics. They represent relationships between two unknown quantities and can be visualized as straight lines on a coordinate plane. When we have two such equations, their intersection point gives us the solution to both equations simultaneously. Today we'll learn how to apply these concepts to solve real-world word problems.
Let's solve a classic age problem step by step. We have two people whose ages sum to 50, and one person is 10 years older than the other. We define x as the younger person's age and y as the older person's age. This gives us two equations: x plus y equals 50, and y equals x plus 10. Substituting the second equation into the first, we get x plus x plus 10 equals 50, which simplifies to 2x equals 40, so x equals 20. Therefore, the younger person is 20 years old and the older person is 30 years old.
Now let's solve a money problem involving coins. Sarah has 15 coins made up of quarters and dimes, with a total value of 2 dollars and 70 cents. We need to find how many of each coin she has. Let x be the number of quarters and y be the number of dimes. Our first equation is x plus y equals 15, representing the total count. Our second equation is 25x plus 10y equals 270, representing the total value in cents. Solving this system, we find that Sarah has 8 quarters and 7 dimes.
Let's explore a mixture problem involving coffee blends. A coffee shop wants to create 20 pounds of a blend worth 6 dollars per pound by mixing coffee that costs 4 dollars per pound with coffee that costs 8 dollars per pound. We need to determine how much of each type to use. Let x be the pounds of 4-dollar coffee and y be the pounds of 8-dollar coffee. Our first equation is x plus y equals 20 for the total weight. Our second equation is 4x plus 8y equals 120 for the total value. Solving this system, we find that they need 10 pounds of each type of coffee.