For a system of 3 linear equations in 3 unknowns ai1x1+ai2x2+ai3x3=bi (i=1,2,3) whose coefficient determinant Δ=a11a21a31a12a22a32a13a23a33 is non-zero, Cramer's rule gives each unknown directly as a ratio of two determinants:
x1=ΔΔ1,x2=ΔΔ2,x3=ΔΔ3,
where Δk is Δ with its kth column replaced by the constants column (b1,b2,b3)T, everything else unchanged. (The same pattern extends to 2 equations in 2 unknowns: Δ=a11a21a12a22, x=Δ1/Δ, y=Δ2/Δ.)
Why it works. Multiplying Δ by x1 and using the linearity-in-a-column property of determinants (splitting the first column ai1x1 into the sum ai1x1+ai2x2+ai3x3 using the original equations, then subtracting off the x2,x3 multiples of the identical columns 2 and 3, which vanish) collapses the first column to exactly the constants bi -- giving x1Δ=Δ1, and dividing by Δ=0 gives the rule.
Worked illustration. For x+y=3,2x−y=0: Δ=121−1=−3, Δ1=301−1=−3, Δ2=1230=−6. So x=Δ1/Δ=1, y=Δ2/Δ=2 -- check: 1+2=3 and 2(1)−2=0, correct.
Word problems that produce equations like y=ax2+bx+c through three given points, or rate/mixture/scoring problems, translate to a 3×3 system in the unknown constants exactly as for matrix inversion, then Cramer's rule reads off each unknown independently -- convenient when only one or two of the unknowns are actually needed. A system with fractional unknowns like xa+by=c is first turned linear by the substitution u=x1 (or y1, z1), solved for u,v,(w) by Cramer's rule, and only inverted back to x,y,(z) at the very last step.
Note
If Δ=0, Cramer's rule simply cannot be applied -- it says nothing about whether the system has no solution or infinitely many; that question needs the rank method (§ Consistency by Rank Method). A closely related determinant trick solves xayb=em,xcyd=en: taking logs turns this into a linear system in lnx,lny, so Cramer's rule on that linear system gives lnx,lny as ratios of 2×2 determinants, and x,y follow by exponentiating.
"Cramer's rule formula for 3 variables" and "Cramer's rule vs matrix method" are frequent search terms among Class 12 students, since this method for solving linear systems is an important topic for JEE Main and several state board and CET-level exams that build on the NCERT Class 12 Determinants and Matrices curriculum. Being able to switch fluently between Cramer's rule and the matrix-inversion method taught in NCERT is a common requirement in exam questions that ask for a specific unknown without solving the full system.
For each part, form D,Dx,Dy,(Dz) and apply x=Dx/D etc.; parts (ii) and (iv) need the substitution u=x1 (and v=y1,w=z1) before Cramer's rule, inverted back at the end.
✓Final answer
x=−2,y=3.
x=21,y=3.
x=2,y=3,z=4.
x=1,y=3,z=3.
Cramer's rule solves a linear system AX=B by xi=Di/D, where D=∣A∣ and Di is D with column i replaced by B. Parts (ii) and (iv) are not linear in x,y,z as printed, so we first substitute u=x1 (and v=y1,w=z1 in (iv)) to make them linear, apply Cramer's rule, then invert back.
Step 1. Part (i): write the system in standard form.5x−2y+16=0⇒5x−2y=−16; x+3y−7=0⇒x+3y=7.
Step 2. Part (i): compute D,Dx,Dy.
D=51−23=5(3)−(−2)(1)=15+2=17
Dx=−167−23=(−16)(3)−(−2)(7)=−48+14=−34
Dy=51−167=5(7)−(−16)(1)=35+16=51
Step 3. Part (i): apply Cramer's rule.x=DDx=17−34=−2,y=DDy=1751=3.
Step 4. Part (ii): substitute u=x1 first. The equations x3+2y=12,x2+3y=13 are not linear in x; with u=x1 they become linear: 3u+2y=12,2u+3y=13.
Step 9. Part (iii): apply Cramer's rule.x=−22−44=2,y=−22−66=3,z=−22−88=4.
Step 10. Part (iv): substitute u=x1,v=y1,w=z1 first. The equations x3−y4−z2=1,x1+y2+z1=2,x2−y5−z4=−1 become linear: 3u−4v−2w=1,u+2v+w=2,2u−5v−4w=−1.
Step 11. Part (iv): compute the main determinant D.