A linear system is two line equations considered together. If it has no solution, the lines never meet: they are parallel but distinct. Rewrite the given equation with both variables on the left, then look for a choice whose variable coefficients scale by the same number but whose constant does not. Scaling the constant too gives the same line, not a separate one.
Hints
- Hint 1
The given equation has on both sides. Subtract the term on the right before comparing coefficients, the numbers multiplying and .
- Hint 2
Fractions can hide a shared multiplier. Multiply every term by a common multiple of the denominators to put the equation in standard form, .
- Hint 3
For no solution, both variable coefficients must scale by the same number, but the constant must not. What constant would you get if you scaled the entire given equation?
Step-by-step
Compare coefficients, then check the constant
Step 1Put both variable terms on the left
The given equation has on both sides. Subtract from both sides:
Now each coefficient, the number multiplying a variable, is ready to compare.
- Step 2
Clear the fractions
Multiply every term by , a common multiple of , , and :
Multiplying both sides by the same nonzero number keeps the same line.
- Step 3
Find the constant for an identical line
To find a same line whose variable coefficients are times those in , multiply the entire equation by :
For no solution, match both variable coefficients but not the constant.
- Step 4
Check that the lines are distinct
That option has , not , as its constant. Type the given equation and on separate Desmos lines. Zoom near the -axis: Desmos shows two parallel lines with a small gap, so they have no shared point. Choice D.