For a linear system, “no solution” means the two lines never meet. They must point in the same direction but be different lines. Scale the given equation to compare its coefficients, then use Desmos to check the graphs. Check whether the lines have the same direction without overlapping. If the constants match too, the lines overlap and share infinitely many solutions.
Hints
- Hint 1
A solution is a point on both lines. No solution means the lines cannot cross. What must be true about their direction, and why can’t they be the very same line?
- Hint 2
Multiply both sides of the given equation by . This clears the fraction and makes its and coefficients easier to compare with the possible second equations.
Step-by-step
Scale the equation and compare the lines
Step 1Graph the line the second equation must avoid
Type in Desmos. It draws the given line. The second line cannot cross it, because a crossing point would solve both equations. Two separate lines that never cross are parallel: they point in the same direction.
- Step 2
Clear the fraction to compare coefficients
A coefficient is the number multiplying a variable. To compare both coefficients with the possible equations, multiply both sides by :
Simplify the products:
- Step 3
Choose a different line with matching coefficients
has the same coefficients as , so the lines point in the same direction. Its constant, the number on the right, is instead of , so they aren’t the same line. Add in Desmos; it shows two separate lines with no crossing. If you used on both right sides, the lines would overlap instead. The second equation is . Choice B.