Chapter 4: Q 259 (page 459)

Use Cramer’s Rule to Solve Systems of Equations
In the following exercises, solve each system of equations using Cramer’s Rule.
$\{\begin{cases} 11 x + 9 y + 2 z = - 9 \\ 7 x + 5 y + 3 z = - 7 \\ 4 x + 3 y + z = - 3 \end{cases}$

Short Answer

Expert verified

The solution for the system of linear equation is, $(2, - 3, - 2)$ .

Step by step solution

Given system of linear equations is,

$\{\begin{cases} 11 x + 9 y + 2 z = - 9 \\ 7 x + 5 y + 3 z = - 7 \\ 4 x + 3 y + z = - 3 \end{cases}$

The objective is, we need to solve this system by using the Cramer's rule.

Find the determinant D by using the coefficients of the variable.

$D = |\begin{matrix} 11 & 9 & 2 \\ 7 & 5 & 3 \\ 4 & 3 & 1 \end{matrix}| = 11 (5 - 9) - 9 (7 - 12) + 2 (21 - 20) = 11 (- 4) - 9 (- 5) + 2 (1) = - 44 + 45 + 2 = 3$

Step 3 Find the determinant Dx by using the constants to replace the coefficients of x.

$D_{x} = |\begin{matrix} - 9 & 9 & 2 \\ - 7 & 5 & 3 \\ - 3 & 3 & 1 \end{matrix}| = - 9 (5 - 9) - 9 (- 7 + 9) + 2 (- 21 + 15) = - 9 (- 4) - 9 (2) + 2 (- 6) = 36 - 18 - 12 = 6$

To find: $x$ ,

$x = \frac{D_{x}}{D} = \frac{6}{3} x = 2$

Step 4 Find the determinant Dy by using the constants to replace the coefficients of y.

$D_{y} = |\begin{matrix} 11 & - 9 & 2 \\ 7 & - 7 & 3 \\ 4 & - 3 & 1 \end{matrix}| = 11 (- 7 + 9) + 9 (7 - 12) + 2 (- 21 + 28) = 11 (2) + 9 (- 5) + 2 (7) = 22 - 45 + 14 = - 9$

To find: $y$ ,

$y = \frac{D_{y}}{D} = \frac{- 9}{3} = - 3$

Step 5 Find the determinants Dz by using the constants to replace the coefficient of z.

$D_{z} = |\begin{matrix} 11 & 9 & - 9 \\ 7 & 5 & - 7 \\ 4 & 3 & - 3 \end{matrix}| = 11 (- 15 + 21) - 9 (- 21 + 28) - 9 (21 - 20) = 11 (6) - 9 (7) - 9 (1) = 66 - 63 - 9 = - 6$

To find: $z$ ,

$z = \frac{D_{z}}{D} = \frac{- 6}{3} = - 2$

The solution for the system of linear equation is, $(2, - 3, - 2)$ .

Step 6 write the solution as an ordered triad.

The ordered triad is, $(x, y, z) = (2, - 3, - 2)$ .

Now substitute $x = 2, y = - 3, z = - 2$ into the equation $7 x + 5 y + 3 z = - 7$ .

$7 (2) + 5 (- 3) + 3 (- 2) = - 7 14 - 15 - 6 = - 7 14 - 21 = - 7 - 7 = - 7$

which is true.

Step 7 Substitute x=2,y=-3,z=-2 into the equation 4x+3y+z=-3.

$4 (2) + 3 (- 3) + (- 2) = - 3 8 - 9 - 2 = - 3 8 - 11 = - 3 - 3 = - 3$

which is true.

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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.
$\{\begin{cases} 11 x + 9 y + 2 z = - 9 \\ 7 x + 5 y + 3 z = - 7 \\ 4 x + 3 y + z = - 3 \end{cases}$

Short Answer

Step by step solution

Given system of linear equations is,

Find the determinant D by using the coefficients of the variable.

Step 3 Find the determinant Dx by using the constants to replace the coefficients of x.

Step 4 Find the determinant Dy by using the constants to replace the coefficients of y.

Step 5 Find the determinants Dz by using the constants to replace the coefficient of z.

Step 6 write the solution as an ordered triad.

Step 7 Substitute x=2,y=-3,z=-2 into the equation 4x+3y+z=-3.

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Use Cramer’s Rule to Solve Systems of EquationsIn the following exercises, solve each system of equations using Cramer’s Rule.11x+9y+2z=−97x+5y+3z=−74x+3y+z=−3

Short Answer

Step by step solution

Given system of linear equations is,

Find the determinant D by using the coefficients of the variable.

Step 3 Find the determinant Dx by using the constants to replace the coefficients of x.

Step 4 Find the determinant Dy by using the constants to replace the coefficients of y.

Step 5 Find the determinants Dz by using the constants to replace the coefficient of z.

Step 6 write the solution as an ordered triad.

Step 7 Substitute x=2,y=-3,z=-2 into the equation 4x+3y+z=-3.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Geometry

Discrete Mathematics

Logic and Functions

Calculus

Theoretical and Mathematical Physics

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.
$\{\begin{cases} 11 x + 9 y + 2 z = - 9 \\ 7 x + 5 y + 3 z = - 7 \\ 4 x + 3 y + z = - 3 \end{cases}$