Q 258 Use Cramer鈥檚 Rule to Solve Sys... [FREE SOLUTION]

91影视

Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 4: Q 258 (page 459)

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

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,

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

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

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

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.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Logic and Functions

Decision Maths

Statistics

Mechanics Maths

Calculus

Study anywhere. Anytime. Across all devices.

91影视

Use Cramer鈥檚 Rule to Solve Systems of EquationsIn the following exercises, solve each system of equations using Cramer鈥檚 Rule.2x鈭�5y+3z=83x鈭�y+4z=7x+3y+2z=鈭�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

Applied Mathematics

Logic and Functions

Decision Maths

Statistics

Mechanics Maths

Calculus

Study anywhere. Anytime. Across all devices.

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