Solve 3x − 2y = −5, −2x + 2y = 2 with Cramer’s rule
Worked out line by line the way a teacher would write it.
Answer
| Solution | x = −3, y = −2 |
Step-by-step solution
5 steps-
1 The system, with the equations numbered\begin{aligned}3 x - 2 y &= -5 & (1)\\ 2 y - 2 x &= 2 & (2)\end{aligned}
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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 & -2 \\ -2 & 2 \end{vmatrix} = 3 \cdot 2 - (-2) \cdot (-2) = 2 -
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} -5 & -2 \\ 2 & 2 \end{vmatrix} = -6 -
4 Replace the y column with the constants to get D_yD_{y} = \begin{vmatrix} 3 & -5 \\ -2 & 2 \end{vmatrix} = -4
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5 Divide each determinant by Dx = \frac{D_{x}}{D} = \frac{-6}{2} = -3,\quad y = \frac{D_{y}}{D} = \frac{-4}{2} = -2
Check by substitution
| Equation | Left side | Right side | |
|---|---|---|---|
| (1) | −5 | −5 | ✓ |
| (2) | 2 | 2 | ✓ |
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