Meet our friends Rabbit and Fox! They went shopping for school supplies - pencils and erasers. But oh no! They forgot to check the individual prices. Looking at their receipts, Rabbit bought 2 pencils and 3 erasers for 13 dollars, while Fox bought 1 pencil and 1 eraser for 5 dollars. Can you help them figure out how much each item costs? This is where systems of equations come to the rescue!
Great! Now let's turn our story into math language. We'll use x for the price of one pencil and y for the price of one eraser. From Rabbit's receipt, we know 2 pencils plus 3 erasers equals 13 dollars, so we write 2x plus 3y equals 13. From Fox's receipt, 1 pencil plus 1 eraser equals 5 dollars, so we write x plus y equals 5. Look! We've created our first system of equations!
Fox has a brilliant idea! 'Let's use the substitution method!' he says. First, we'll solve the simpler equation for one variable. From x plus y equals 5, we can isolate y to get y equals 5 minus x. Now comes the clever part - we substitute this expression for y into our first equation! So 2x plus 3y equals 13 becomes 2x plus 3 times parentheses 5 minus x parentheses equals 13. Rabbit looks a bit confused, but Fox is excited about this mathematical magic!
Now for the exciting part - solving our equation! Let's work through 2x plus 3 times parentheses 5 minus x parentheses equals 13. First, we distribute: 2x plus 15 minus 3x equals 13. Combining like terms: negative x plus 15 equals 13. Subtracting 15 from both sides: negative x equals negative 2. So x equals 2! That means each pencil costs 2 dollars. Now we can find y: y equals 5 minus 2, which equals 3. Each eraser costs 3 dollars! Let's double-check: 2 times 2 plus 3 times 3 equals 4 plus 9 equals 13. Check! And 2 plus 3 equals 5. Perfect! Our animal friends are so happy they solved the mystery!
Here comes the most amazing part! Each equation actually represents a straight line on a coordinate plane. The red line shows all points where 2x plus 3y equals 13, and the blue line shows all points where x plus y equals 5. When we graph both lines, they intersect at exactly one point - coordinates 2 comma 3! This intersection point is our solution because it's the only point that satisfies both equations at the same time. This is the beautiful geometric meaning of systems of equations - finding where two lines meet! Rabbit and Fox are so excited they've discovered this mathematical magic!