Consistency Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
For what value of kk is {x+2y=32x+4y=k\begin{cases} x + 2y = 3 \\ 2x + 4y = k \end{cases} consistent?

Solution

  1. 1
    Step 1: Equation 2 is 2ร—2 \times equation 1 if k=6k = 6.
  2. 2
    Step 2: If k=6k = 6: dependent (infinitely many solutions) โ€” consistent.
  3. 3
    Step 3: If kโ‰ 6k \neq 6: parallel lines โ€” inconsistent.

Answer

k=6k = 6
When the coefficient ratios match (12=24\frac{1}{2} = \frac{2}{4}), consistency requires the constant ratio to also match: 3k=12\frac{3}{k} = \frac{1}{2}, giving k=6k = 6.

About Consistency

A system of equations is consistent when there exists at least one set of variable values that satisfies every equation simultaneously.

Learn more about Consistency โ†’

More Consistency Examples

Example 1 easy

Is the system [formula] consistent?

Example 3 easy

Is [formula] consistent?

Example 4 medium

Is [formula] consistent?

โ† All Consistency Examples Consistency Concept