Solve x + y = 1, −5x − y = −25 with Cramer’s rule
Worked out line by line the way a teacher would write it.
Answer
| Solution | x = 6, y = −5 |
Step-by-step solution
5 steps-
1 The system, with the equations numbered\begin{aligned}x + y &= 1 & (1)\\ - 5 x - y &= -25 & (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 \\ -5 & -1 \end{vmatrix} = 1 \cdot (-1) - 1 \cdot (-5) = 4 -
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} 1 & 1 \\ -25 & -1 \end{vmatrix} = 24 -
4 Replace the y column with the constants to get D_yD_{y} = \begin{vmatrix} 1 & 1 \\ -5 & -25 \end{vmatrix} = -20
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5 Divide each determinant by Dx = \frac{D_{x}}{D} = \frac{24}{4} = 6,\quad y = \frac{D_{y}}{D} = \frac{-20}{4} = -5
Check by substitution
| Equation | Left side | Right side | |
|---|---|---|---|
| (1) | 1 | 1 | ✓ |
| (2) | −25 | −25 | ✓ |
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