Solve −x − y = −5, 3x + 5y = 29 with Cramer’s rule
Worked out line by line the way a teacher would write it.
Answer
| Solution | x = −2, y = 7 |
Step-by-step solution
5 steps-
1 The system, with the equations numbered\begin{aligned}- x - y &= -5 & (1)\\ 3 x + 5 y &= 29 & (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} -1 & -1 \\ 3 & 5 \end{vmatrix} = (-1) \cdot 5 - (-1) \cdot 3 = -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 & -1 \\ 29 & 5 \end{vmatrix} = 4 -
4 Replace the y column with the constants to get D_yD_{y} = \begin{vmatrix} -1 & -5 \\ 3 & 29 \end{vmatrix} = -14
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5 Divide each determinant by Dx = \frac{D_{x}}{D} = \frac{4}{-2} = -2,\quad y = \frac{D_{y}}{D} = \frac{-14}{-2} = 7
Check by substitution
| Equation | Left side | Right side | |
|---|---|---|---|
| (1) | −5 | −5 | ✓ |
| (2) | 29 | 29 | ✓ |
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