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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 262 (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.
$\{\begin{cases} 2 y + 3 z = 鈭� 1 \\ 5 x + 3 y = 鈭� 6 \\ 7 x + z = 1 \end{cases}$

Short Answer

Expert verified

The solution for the system of linear equation is, $(0, - 2, 1)$ .

Step by step solution

01

Step 1Given system of linear equation is,

$\{\begin{cases} 2 y + 3 z = 鈭� 1 \\ 5 x + 3 y = 鈭� 6 \\ 7 x + z = 1 \end{cases}$

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

02

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

$D = |\begin{matrix} 0 & 2 & 3 \\ 5 & 3 & 0 \\ 7 & 0 & 1 \end{matrix}| = 0 - 2 (5 - 0) + 3 (0 - 21) = 0 - 2 (5) + 3 (- 21) = - 10 - 63 = - 73$

03

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

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

To find: $x$

$x = \frac{D_{x}}{D} = \frac{0}{- 73} = 0$

04

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

$D_{y} = |\begin{matrix} 0 & - 1 & 3 \\ 5 & - 6 & 0 \\ 7 & 1 & 1 \end{matrix}| = 0 - (- 1) (5 - 0) + 3 (5 + 42) = 5 + 3 (47) = 146$

To find: $y$

$y = \frac{D_{y}}{D} = \frac{146}{- 73} y = - 2$

05

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

$D_{z} = |\begin{matrix} 0 & 2 & - 1 \\ 5 & 3 & - 6 \\ 7 & 0 & 1 \end{matrix}| = 0 - 2 (5 + 42) - 1 (0 - 21) = - 2 (47) - 1 (- 21) = - 94 + 21 = - 73$

To find: $z$

$z = \frac{D_{z}}{D} = \frac{- 73}{- 73} z = 1$

The solution for the system of linear equation is, $(0, - 2, 1)$ .

06

Step 6 write the solution as an ordered triad.

The ordered triad is, $(x, y, z) = (0, - 2, 1)$ .

Substitute $x = 0, y = - 2, z = 1$ into the equation $2 y + 3 z = - 1$ .

$2 (- 2) + 3 (1) = - 1 - 4 + 3 = - 1 - 1 = - 1$

which is true.

07

Step 7 substitute x=0,y=-2,z=1 into the equation 7x+z=1.

$7 (0) + 1 = 1 0 + 1 = 1 1 = 1$

which is true.

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Most popular questions from this chapter

A commercial jet can fly $1320$ miles in $3$ hours with a tailwind but only $1170$ miles in $3$ hours into a headwind. Find the speed of the jet in still air and the speed of the wind.

Solve the system by elimination: $\{\begin{cases} 4 x + y = - 5 \\ - 2 x - 2 y = - 2 \end{cases}$

Solve the system of equations by substitution

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