Question

A Curious System
Andre is trying to solve this system of equations:
x + y = 3
4x = 12 – 4y
Looking at the first equation, he thought, "The solution to the system is a pair of numbers that add
up to 3. I wonder which two numbers they are."
1. Choose any two numbers that add up to 3. Let the first one be the x-value and the second
one be the y-value.
2. The pair of values you chose is a solution to the first equation. Check if it is also a solution
to the second equation. Then, pause for a brief discussion with your group. Did your
classmates’ values also work in the second equation?
3. How many solutions does the system have? Be prepared to defend your answer.

Answer 1

Pick any two numbers that add to 3. For example, choose x = 1 and y = 2 (since 1 + 2 = 3).

Check the second equation:
4x = 4(1) = 4
12 − 4y = 12 − 4(2) = 12 − 8 = 4
So 4x = 12 − 4y holds, so (1, 2) satisfies both equations.

Algebraic check for all such pairs:
From x + y = 3 we have y = 3 − x. Substitute in the second equation:
4x = 12 − 4y = 12 − 4(3 − x) = 12 − 12 + 4x = 4x,
which is always true. Equivalently, rewrite the second as 4x + 4y = 12, then divide by 4 to get x + y = 3. The two equations are the same.

Conclusion:
The system has infinitely many solutions — every pair (x, y) with y = 3 − x (i.e., every point on the line x + y = 3). Any classmates’ pair that adds to 3 will also satisfy the second equation.