Solve a System with Infinitely Many Solutions

Problem

2x + y = 6,\quad 4x + 2y = 12

Answer

\text{infinitely many solutions}

Sometimes the two equations in a system turn out to be the same line. Then every point on the line is a solution, and the system has infinitely many of them.

Step-by-step solution

Multiply the first equation by 2.
$4x + 2y = 12$
Why: Matching the coefficients makes the two equations easy to compare.
Compare with the second equation.
$4x + 2y = 12 \text{ and } 4x + 2y = 12$
Why: After the multiplication, both equations are identical.
Name the result.
$\text{infinitely many solutions}$
Why: Every point on the line 2x + y = 6 solves both equations, so there is no single pair.

Common mistakes

Writing one number as the answer. A dependent system has no single solution.
Saying no solution. Identical equations share every point.
Stopping before comparing the equations, which hides the match.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

$x + y = 3,\quad 2x + 2y = 6$

Show answer

$\text{infinitely many solutions}$
$x - y = 1,\quad 3x - 3y = 3$

Show answer

$\text{infinitely many solutions}$
$2x + y = 4,\quad 2x + y = 5$

Show answer

$\text{no solution}$
$x + 2y = 4,\quad 2x + 4y = 8$

Show answer

$\text{infinitely many solutions}$
$3x + y = 2,\quad 6x + 2y = 4$

Show answer

$\text{infinitely many solutions}$