Chapter 4: Q 258 (page 459)

Use Cramer’s Rule to Solve Systems of Equations
In the following exercises, solve each system of equations using Cramer’s Rule.
$\{\begin{cases} 2 x - 5 y + 3 z = 8 \\ 3 x - y + 4 z = 7 \\ x + 3 y + 2 z = - 3 \end{cases}$

Short Answer

Expert verified

The solution for the system of linear equation is, $(6, - 1, - 3)$ .

Step by step solution

Given system of linear equation is,

$\{\begin{cases} 2 x - 5 y + 3 z = 8 \\ 3 x - y + 4 z = 7 \\ x + 3 y + 2 z = - 3 \end{cases}$

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

Step 2 Find the determinant D by using the coefficients of the variable.

$D = |\begin{matrix} 2 & - 5 & 3 \\ 3 & - 1 & 4 \\ 1 & 3 & 2 \end{matrix}| = 2 (- 2 - 12) + 5 (6 - 4) + 3 (9 + 1) = 2 (- 14) + 5 (2) + 3 (10) = - 28 + 10 + 30 = 12$

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

$D_{x} = |\begin{matrix} 8 & - 5 & 3 \\ 7 & - 1 & 4 \\ - 3 & 3 & 2 \end{matrix}| = 8 (- 2 - 12) + 5 (14 + 12) + 3 (21 - 3) = 8 (- 14) + 5 (26) + 3 (18) = - 112 + 130 + 54 = 72$

To find: $x$ ,

$x = \frac{D_{x}}{D} = \frac{72}{12} = 6$

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

$D_{y} = |\begin{matrix} 2 & 8 & 3 \\ 3 & 7 & 4 \\ 1 & - 3 & 2 \end{matrix}| = 2 (14 + 12) - 8 (6 - 4) + 3 (- 9 - 7) = 2 (26) - 8 (2) + 3 (- 16) = 52 - 16 - 48 = - 12$

To find: $y$ ,

$y = \frac{D_{y}}{D} = \frac{- 12}{12} = - 1$

Step 5 Find the determinant Dz by using the constants to replace the coefficients of z.

$D_{z} = |\begin{matrix} 2 & - 5 & 8 \\ 3 & - 1 & 7 \\ 1 & 3 & - 3 \end{matrix}| = 2 (3 - 21) + 5 (- 9 - 7) + 8 (9 + 1) = 2 (- 18) + 5 (- 16) + 8 (10) = - 36 - 80 + 80 = - 36$

To find: $z$ ,

$z = \frac{D_{z}}{D} = \frac{- 36}{12} = - 3$

The solution for the system of linear equation is, $(6, - 1, - 3)$ .

Step 6 write the solution as an ordered triad.

The ordered triad is, $(x, y, z) = (6, - 1, - 3)$ .

Now substitute $x = 6, y = - 1, z = - 3$ into the equation $2 x - 5 y + 3 z = 8$ .

role="math" localid="1644735602841" $2 x - 5 y + 3 z = 8 2 (6) - 5 (- 1) + 3 (- 3) = 8 12 + 5 - 9 = 8 12 - 4 = 8 8 = 8$

which is true.

Step 7 Substitute x=6,y=-1,z=-3 into the equation 3x-y+4z=7.

$3 (6) - (- 1) + 4 (- 3) = 7 18 + 1 - 12 = 7 7 = 7$

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.
$\{\begin{cases} 2 x - 5 y + 3 z = 8 \\ 3 x - y + 4 z = 7 \\ x + 3 y + 2 z = - 3 \end{cases}$

Short Answer

Step by step solution

Given system of linear equation is,

Step 2 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 determinants Dy by using the constants to replace the coefficients of y.

Step 5 Find the determinant Dz by using the constants to replace the coefficients of z.

Step 6 write the solution as an ordered triad.

Step 7 Substitute x=6,y=-1,z=-3 into the equation 3x-y+4z=7.

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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.2x−5y+3z=83x−y+4z=7x+3y+2z=−3

Short Answer

Step by step solution

Given system of linear equation is,

Step 2 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 determinants Dy by using the constants to replace the coefficients of y.

Step 5 Find the determinant Dz by using the constants to replace the coefficients of z.

Step 6 write the solution as an ordered triad.

Step 7 Substitute x=6,y=-1,z=-3 into the equation 3x-y+4z=7.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Decision Maths

Discrete Mathematics

Calculus

Mechanics Maths

Pure 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.
$\{\begin{cases} 2 x - 5 y + 3 z = 8 \\ 3 x - y + 4 z = 7 \\ x + 3 y + 2 z = - 3 \end{cases}$