Chapter 4: Q 257 (page 459)

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

Short Answer

Expert verified

The solution for the system of linear equation is, $(- 3, - 5, 4)$ .

Step by step solution

Given system of linear equation is,

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

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

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

$D = |\begin{matrix} 4 & - 3 & 1 \\ 2 & - 5 & - 4 \\ 3 & - 2 & - 2 \end{matrix}| = 4 (10 - 8) + 3 (- 4 + 12) + 1 (- 4 + 15) = 4 (2) + 3 (8) + 1 (11) = 8 + 24 + 11 = 43$

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

$D_{x} = |\begin{matrix} 7 & - 3 & 1 \\ 3 & - 5 & - 4 \\ - 7 & - 2 & - 2 \end{matrix}| = 7 (10 - 8) + 3 (- 6 - 28) + 1 (- 6 - 35) = 7 (2) + 3 (- 34) + 1 (- 41) = 14 - 102 - 41 = - 129$

To find: $x$ ,

$x = \frac{D_{x}}{D} = \frac{- 129}{43} = - 3$

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

$D_{y} = |\begin{matrix} 4 & 7 & 1 \\ 2 & 3 & - 4 \\ 3 & - 7 & - 2 \end{matrix}| = 4 (- 6 - 28) - 7 (- 4 + 12) + 1 (- 14 - 9) = 4 (- 34) - 7 (8) + 1 (- 23) = - 136 - 56 - 23 = - 215$

To find: $y$ ,

$y = \frac{D_{y}}{D} = \frac{- 215}{43} = - 5$

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

$D_{z} = |\begin{matrix} 4 & - 3 & 7 \\ 2 & - 5 & 3 \\ 3 & - 2 & - 7 \end{matrix}| = 4 (35 + 6) + 3 (- 14 - 9) + 7 (- 4 + 15) = 4 (41) + 3 (- 23) + 7 (11) = 164 - 69 + 77 = 172$

To find: $z$ ,

$z = \frac{D_{z}}{D} = \frac{172}{43} = 4$

The solution for the system of linear equation is, $(- 3, - 5, 4)$ .

Step 6 write the solution as an ordered triad.

The ordered triad is, $(x, y, z) = (- 3, - 5, 4)$ .

Substitute $x = - 3, y = - 5, z = 4$ into the equation $4 x - 3 y + z = 7$ .

$4 (- 3) - 3 (- 5) + 4 = 7 - 12 + 15 + 4 = 7 3 + 4 = 7 7 = 7$

which is true.

Step 7 Now substitute x=-3,y=-5,z=4 into the equation 2x-5y-4z=3.

$2 (- 3) - 5 (- 5) - 4 (4) = 3 - 6 + 25 - 16 = 3 25 - 22 = 3 3 = 3$

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

Short Answer

Step by step solution

Given system of linear equation is,

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

Step 3 Find the determinant Dxby 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 determinant Dz by using the constants to replace the coefficients of z.

Step 6 write the solution as an ordered triad.

Step 7 Now substitute x=-3,y=-5,z=4 into the equation 2x-5y-4z=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.4x−3y+z=72x−5y−4z=33x−2y−2z=−7

Short Answer

Step by step solution

Given system of linear equation is,

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

Step 3 Find the determinant Dxby 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 determinant Dz by using the constants to replace the coefficients of z.

Step 6 write the solution as an ordered triad.

Step 7 Now substitute x=-3,y=-5,z=4 into the equation 2x-5y-4z=3.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Logic and Functions

Theoretical and Mathematical Physics

Probability and Statistics

Statistics

Calculus

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