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Solve each system of equations using a matrix.

2y+3z=-15x+3y=-67x+z=1

Short Answer

Expert verified

The solution for the system of linear equation is0,-2,1.

Step by step solution

01

Step 1. Given information.

Consider the given system of equations,

2y+3z=-15x+3y=-67x+z=1

02

Step 2. Write in augmented form.

The augmented matrix for the given system of equations is

023-1530-67011

03

Step 3. Apply row operations.

Interchanging rowsR1andR2,

530-6023-17011

Apply R15โ†’R1and R3-7ร—R1โ†’R3,

localid="1644431494537" 1350-65023-10-2151475

Apply R22โ†’R2and R3+215ร—R2โ†’R3,

localid="1644431562941" 1350-650132-120073107310

Apply R3ร—1073โ†’R3,

1350-650132-120011

Now, the matrix is in row-echelon form.

04

Step 4. Write in system of equations.

Writing the corresponding system of equations,

x+35y=-65......(i)y+32z=-12......(ii)z=1......(iii)

Substitute z=1in equation (ii),

y+32ร—1=-12y=-32-12y=-42y=-2

Substitute y=-2in equation (i),

x+35ร—-2=-65x-65=-65x=0

05

Step 5. Check the answers.

Substitute the values x,yin equation (i),

0+35-2=-650-65=-65-65=-65

This is true.

Substitute the values y,zin equation (ii),

role="math" localid="1644591135020" -2+321=-12-2+32=-12-12=-12

This is also true.

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