Solve −3x − 3y = −9, −2x + 4y = −24 with Cramer’s rule
Worked out line by line the way a teacher would write it.
Answer
| Solution | x = 6, y = −3 |
Step-by-step solution
5 steps-
1 The system, with the equations numbered\begin{aligned}- 3 x - 3 y &= -9 & (1)\\ 4 y - 2 x &= -24 & (2)\end{aligned}
-
2 Compute the determinant D of the coefficient matrix
Cramer’s rule needs the determinant D of the coefficient matrix. When D ≠ 0 the system has exactly one solution.
D = \begin{vmatrix} -3 & -3 \\ -2 & 4 \end{vmatrix} = (-3) \cdot 4 - (-3) \cdot (-2) = -18 -
3 Replace the x column with the constants to get D_x
Replace one unknown’s column with the constants and take the determinant: dividing it by D gives that unknown, for example x = D_x / D. Each unknown gets its own determinant this way.
D_{x} = \begin{vmatrix} -9 & -3 \\ -24 & 4 \end{vmatrix} = -108 -
4 Replace the y column with the constants to get D_yD_{y} = \begin{vmatrix} -3 & -9 \\ -2 & -24 \end{vmatrix} = 54
-
5 Divide each determinant by Dx = \frac{D_{x}}{D} = \frac{-108}{-18} = 6,\quad y = \frac{D_{y}}{D} = \frac{54}{-18} = -3
Check by substitution
| Equation | Left side | Right side | |
|---|---|---|---|
| (1) | −9 | −9 | ✓ |
| (2) | −24 | −24 | ✓ |
Open this problem in the solver
Change a number or try your own problem: the solver works it out the same way, with a graph and hint mode.