Consider the pair of linear equations below.

Question

4x+6y=12
2x+3y=6

Part A: What is the relationship, if any, between 12 and 6?

Part B: Do the two equations have one solution, no solution, or infinitely many solution? Explain.

Part C: How can you verify your answers to Part A and B by solving algebraically?

Answer 1

Eq1: 4x + 6y = 12.
m1 = -A/B = -4/6 = -2/3.
Y-int. = C/B = 12/6 = 2.

Eq2: 2x + 3y = 6.
m2 = -2/3.
Y-int. = 6/3 = 2.

Since the slopes and Y-intercepts are both equal, the Eqs represent the same line. Therefore, we have an infinite number of solutions.

Multiply both sides of Eq2 by 2 and compare the Eqs.

If the slopes were not equal, there will be one solution.

If the slopes were equal with unequal Y-intercepts, we'll have no solutions.
In this case, we have two parallel lines.