The system of linear equations 3x+y+4=0 and 2x+y+9=0 has:

The correct answer is Option (2) → Unique Solution

Two linear equations \(a_{1}x+b_{1}y+c_{1}=0\) and \(a_{2}x+b_{2}y+c_{2}=0\) have a unique solution when \(\frac{a_{1}}{a_{2}}\ne\frac{b_{1}}{b_{2}}\): the two lines have different slopes, so they meet at one point.

• \(\displaystyle\frac{a_{1}}{a_{2}}=\frac{3}{2}\) and \(\displaystyle\frac{b_{1}}{b_{2}}=\frac{1}{1}=1\)

• \(\frac{3}{2}\ne1\), so there is one and only one solution

Subtracting the second equation from the first, \(y\) cancels: \(x-5=0\), so \(x=5\), and then \(y=-19\).

Tip: Compare the ratios of the coefficients: \(\frac{a_{1}}{a_{2}}\ne\frac{b_{1}}{b_{2}}\) gives one solution; \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}\ne\frac{c_{1}}{c_{2}}\) gives none (parallel lines); all three ratios equal gives infinitely many (the same line).