Chapter 4: Q 271 (page 459)

Use Cramer’s Rule to Solve Systems of Equations In the following exercises, solve each system of equations using Cramer’s Rule.
$\{\begin{array}{l} x - 2 y + 3 z = 1 \\ x + y - 3 z = 7 \\ 3 x - 4 y + 5 z = 7 \end{array}$

Short Answer

Expert verified

The system is consistent and dependent and it has infinitely many solutions.

Step by step solution

Given system of equations are,

$\{\begin{cases} x - 2 y + 3 z = 1 \\ x + y - 3 z = 7 \\ 3 x - 4 y + 5 z = 7 \end{cases}$

To find:

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

Step 3 First find the determinant D by using the coefficients of the variables.

$D = |\begin{matrix} 1 & - 2 & 3 \\ 1 & 1 & - 3 \\ 3 & - 4 & 5 \end{matrix}| D = 1 (5 - 12) + 2 (5 + 9) + 3 (- 4 - 3) = 1 (- 7) + 2 (14) + 3 (- 7) = - 7 + 28 - 21 = 0$

Here we cannot use Cramer's rule. But by looking at the value of the determinants $D_{x}, D_{y}, D_{z}$ . Now 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.

$D_{x} = |\begin{matrix} 1 & - 2 & 3 \\ 7 & 1 & - 3 \\ 7 & - 4 & 5 \end{matrix}| = 1 (5 - 12) + 2 (35 + 21) + 3 (- 28 - 7) = 1 (- 7) + 2 (56) + 3 (- 35) = - 7 + 112 - 105 = 0$

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

$D_{y} = |\begin{matrix} 1 & 1 & 3 \\ 1 & 7 & - 3 \\ 3 & 7 & 5 \end{matrix}| = 1 (35 + 21) - 1 (5 + 9) + 3 (7 - 21) = 1 (56) - 1 (14) + 3 (- 14) = 56 - 14 - 42 = 0$

Step 6 Evaluate the determinant Dz, use the constants to replace the coefficients of z.

$D_{z} = |\begin{matrix} 1 & - 2 & 1 \\ 1 & 1 & 7 \\ 3 & - 4 & 7 \end{matrix}| = 1 (7 + 28) + 2 (7 - 21) + 1 (- 4 - 3) = 1 (35) + 2 (- 14) + 1 (- 7) = 35 - 28 - 7 = 0$

Here, all the determinants are equal to zero.

So, the system is consistent and dependent and it has infinitely many solutions.

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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.
$\{\begin{array}{l} x - 2 y + 3 z = 1 \\ x + y - 3 z = 7 \\ 3 x - 4 y + 5 z = 7 \end{array}$

Short Answer

Step by step solution

Given system of equations are,

To find:

Step 3 First 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 Now evaluate the determinant Dy, use the constants to replace the coefficients of y.

Step 6 Evaluate the determinant Dz, use 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.x-2y+3z=1x+y-3z=73x-4y+5z=7

Short Answer

Step by step solution

Given system of equations are,

To find:

Step 3 First 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 Now evaluate the determinant Dy, use the constants to replace the coefficients of y.

Step 6 Evaluate the determinant Dz, use 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

Applied Mathematics

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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.
$\{\begin{array}{l} x - 2 y + 3 z = 1 \\ x + y - 3 z = 7 \\ 3 x - 4 y + 5 z = 7 \end{array}$