Chapter 4: Q 269 (page 459)

Use Cramer’s Rule to Solve Systems of Equations.
In the following exercises, solve each system of equations using Cramer’s Rule.
$\{\begin{cases} 2 x + 3 y + z = 12 \\ x + y + z = 9 \\ 3 x + 4 y + 2 z = 20 \end{cases}$

Short Answer

Expert verified

The system is inconsistent and it has no solution.

Step by step solution

Given system of equations are,

$\{\begin{cases} 2 x + 3 y + z = 12 \\ x + y + z = 9 \\ 3 x + 4 y + 2 z = 20 \end{cases}$

to find:

We need to find the solution for the system of equations by using the Cramer's rule.

Step 3 Find the determinant D by using the coefficients of the variables.

$D = |\begin{matrix} 2 & 3 & 1 \\ 1 & 1 & 1 \\ 3 & 4 & 2 \end{matrix}| = 2 (2 - 4) - 3 (2 - 3) + 1 (4 - 3) = 2 (- 2) - 3 (- 1) + 1 (1) = - 4 + 3 + 1 = 0$

We cannot use the Cramer's rule to solve this system.

But by looking at the value of the determinants $D_{x}, D_{y}, D_{z}$ , we can determine whether the system is dependent or inconsistent.

Step 4 Evaluate the determinant Dx, use the constants to replace the coefficients of x.

$D_{x} = |\begin{matrix} 12 & 3 & 1 \\ 9 & 1 & 1 \\ 20 & 4 & 2 \end{matrix}| = 12 (2 - 4) - 3 (18 - 20) + 1 (36 - 20) = 12 (- 2) - 3 (- 2) + 1 (16) = - 24 + 6 + 16 = - 2$

Step 5 Evaluate the determinant Dy,use the constants to replace the coefficients of y.

$D_{y} = |\begin{matrix} 2 & 12 & 1 \\ 1 & 9 & 1 \\ 3 & 20 & 2 \end{matrix}| = 2 (18 - 20) - 12 (2 - 3) + 1 (20 - 27) = 2 (- 2) - 12 (- 1) + 1 (- 7) = - 4 + 12 - 7 = 1$

Step 6 Evaluating the determinant Dz, by using the constants to replace the coefficients of z.

$D_{z} = |\begin{matrix} 2 & 3 & 12 \\ 1 & 1 & 9 \\ 3 & 4 & 20 \end{matrix}| = 2 (20 - 36) - 3 (20 - 27) + 12 (4 - 3) = 2 (- 16) - 3 (- 7) + 12 (1) = - 32 + 21 + 12 = 1$

All the determinants are not equal to zero.

So, the system is inconsistent and it has no solution.

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Use Cramer’s Rule to Solve Systems of Equations.
In the following exercises, solve each system of equations using Cramer’s Rule.
$\{\begin{cases} 2 x + 3 y + z = 12 \\ x + y + z = 9 \\ 3 x + 4 y + 2 z = 20 \end{cases}$

Short Answer

Step by step solution

Given system of equations are,

to find:

Step 3 Find the determinant D by using the coefficients of the variables.

Step 4 Evaluate the determinant Dx, use the constants to replace the coefficients of x.

Step 5 Evaluate the determinant Dy,use the constants to replace the coefficients of y.

Step 6 Evaluating the determinant Dz, by using the constants to replace the coefficients of z.

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Use Cramer’s Rule to Solve Systems of Equations.In the following exercises, solve each system of equations using Cramer’s Rule.2x+3y+z=12x+y+z=93x+4y+2z=20

Short Answer

Step by step solution

Given system of equations are,

to find:

Step 3 Find the determinant D by using the coefficients of the variables.

Step 4 Evaluate the determinant Dx, use the constants to replace the coefficients of x.

Step 5 Evaluate the determinant Dy,use the constants to replace the coefficients of y.

Step 6 Evaluating the determinant Dz, by using the constants to replace the coefficients of z.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Geometry

Pure Maths

Theoretical and Mathematical Physics

Decision Maths

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Use Cramer’s Rule to Solve Systems of Equations.
In the following exercises, solve each system of equations using Cramer’s Rule.
$\{\begin{cases} 2 x + 3 y + z = 12 \\ x + y + z = 9 \\ 3 x + 4 y + 2 z = 20 \end{cases}$