Solve 4x − 2y = 0, 3x − 4y = −10 with Cramer’s rule
Worked out line by line the way a teacher would write it.
Answer
| Solution | x = 2, y = 4 |
Step-by-step solution
5 steps-
1 The system, with the equations numbered\begin{aligned}4 x - 2 y &= 0 & (1)\\ 3 x - 4 y &= -10 & (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} 4 & -2 \\ 3 & -4 \end{vmatrix} = 4 \cdot (-4) - (-2) \cdot 3 = -10 -
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} 0 & -2 \\ -10 & -4 \end{vmatrix} = -20 -
4 Replace the y column with the constants to get D_yD_{y} = \begin{vmatrix} 4 & 0 \\ 3 & -10 \end{vmatrix} = -40
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5 Divide each determinant by Dx = \frac{D_{x}}{D} = \frac{-20}{-10} = 2,\quad y = \frac{D_{y}}{D} = \frac{-40}{-10} = 4
Check by substitution
| Equation | Left side | Right side | |
|---|---|---|---|
| (1) | 0 | 0 | ✓ |
| (2) | −10 | −10 | ✓ |
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