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QuestionGiải hệ phương trình: −2mx+y=5-2mx + y = 5 và mx+3y=1mx + 3y = 1. Tìm nghiệm khi m=1m=1 và giá trị mm để hệ có nghiệm duy nhất.

Studdy Solution

STEP 1

Assumptions1. We have a system of two linear equations with two variables x and y. . The coefficients of x and y in these equations are dependent on a parameter m.
3. We are asked to solve the system for a specific value of m (m =1).
4. We are also asked to determine the value of m for which the system has a unique solution, and to find that solution.

STEP 2

First, let's solve the system of equations when m =1. The system becomes{−2x+y=5x+y=1\left\{\begin{array}{c}-2 x+y=5 \\ x+ y=1\end{array}\right.

STEP 3

We can solve this system by substitution or elimination. Here, we will use substitution. First, let's express y from the first equationy=2x+5y =2x +5

STEP 4

Now, substitute y in the second equationx+3(2x+)=1x +3(2x +) =1

STEP 5

implify the equation to find the value of xx+x+15=1x +x +15 =17x=−147x = -14x=−2x = -2

STEP 6

Substitute x = -2 into the first equation to find the value of y−2(−2)+y=5-2(-2) + y =5y=5−4y =5 -4y=1y =1So, when m =1, the solution of the system is x = -2, y =1.

STEP 7

Now, let's find the value of m for which the system has a unique solution. A system of two linear equations has a unique solution if and only if the determinant of the coefficient matrix is not zero.
The coefficient matrix of the system is(−2m1m3)\left(\begin{array}{cc}-2m &1 \\ m &3\end{array}\right)

STEP 8

Calculate the determinant of the coefficient matrix=−2m⋅3−1⋅m=−6m−m=−7m = -2m \cdot3 -1 \cdot m = -6m - m = -7m

STEP 9

Set the determinant equal to zero and solve for m−7m=-7m =m=m =So, the system has a unique solution when m ≠.

STEP 10

Now, let's find the unique solution of the system for m ≠0. We can do this by using Cramer's rule.
The determinant of the coefficient matrix is -7m. The determinant of the matrix obtained by replacing the first column of the coefficient matrix with the column of constants isx=∣53∣=5⋅3−⋅=14_x = \left|\begin{array}{cc}5 & \\ &3\end{array}\right| =5 \cdot3 - \cdot =14The determinant of the matrix obtained by replacing the second column of the coefficient matrix with the column of constants isy=∣−2m5m∣=−2m⋅−5⋅m=−7m_y = \left|\begin{array}{cc}-2m &5 \\ m &\end{array}\right| = -2m \cdot -5 \cdot m = -7m

STEP 11

Now, calculate the values of x and y using Cramer's rulex=Dx/D=14/−7m=−/mx = D_x / D =14 / -7m = -/my=Dy/D=−7m/−7m=y = D_y / D = -7m / -7m =So, when m ≠0, the unique solution of the system is x = -/m, y =.

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