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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 4: Q 261 (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.
$\{\begin{cases} 2 x + 5 y = 4 \\ 3 y 鈭� z = 3 \\ 4 x + 3 z = 鈭� 3 \end{cases}$

Short Answer

Expert verified

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

Step by step solution

01

Given system of linear equation is,

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

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

02

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

$D = |\begin{matrix} 2 & 5 & 0 \\ 0 & 3 & - 1 \\ 4 & 0 & 3 \end{matrix}| = 2 (9 - 0) - 5 (0 + 4) + 0 = 18 - 20 = - 2$

03

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

$D_{x} = |\begin{matrix} 4 & 5 & 0 \\ 3 & 3 & - 1 \\ - 3 & 0 & 3 \end{matrix}| = 4 (9 - 0) - 5 (9 - 3) + 0 = 4 (9) - 5 (6) = 36 - 30 = 6$

To find: $x$ ,

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

04

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

$D_{y} = |\begin{matrix} 2 & 4 & 0 \\ 0 & 3 & - 1 \\ 4 & - 3 & 3 \end{matrix}| = 2 (9 - 3) - 4 (0 + 4) + 0 = 2 (6) - 4 (4) = 12 - 16 = - 4$

To find: $y$ ,

$y = \frac{D_{y}}{D} = \frac{- 4}{- 2} = 2$

05

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

$D_{z} = |\begin{matrix} 2 & 5 & 4 \\ 0 & 3 & 3 \\ 4 & 0 & - 3 \end{matrix}| = 2 (- 9 - 0) - 5 (0 - 12) + 4 (0 - 12) = 2 (- 9) - 5 (- 12) + 4 (- 12) = - 18 + 60 - 48 = - 6$

To find: $z$ ,

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

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

06

Step 6 Write the solution as an ordered triad.

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

Substitute $x = - 3, y = 2, z = 3$ into the equation $2 x + 5 y = 4$ .

$2 (- 3) + 5 (2) = 4 - 6 + 10 = 4 4 = 4$

which is true.

07

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

$3 (2) - (3) = 3 6 - 3 = 3 3 = 3$

which is true.

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Most popular questions from this chapter

The sum of two numbers is 10 One number is 4 less than the other. Find the numbers.

Solve the system by substitution:

\{\begin{cases} 4 x - y = 0 \\ 2 x - 3 y = 5 \end{cases}

Solve the system by elimination: $\{\begin{cases} 3 x + y = 5 \\ 2 x - 3 y = 7 \end{cases}$

Solve the system by graphing: $\{\begin{cases} - x + y = 1 \\ 3 x + 2 y = 12 \end{cases}$

The sum of two number is $15$ . One number is $3$ less than the other. Find the numbers.

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