Matrix With Infinitely Many Solutions

Learning Objectives

By the finish of this department, you lot volition be able to:

• Write the augmented matrix for a system of equations
• Apply row operations on a matrix
• Solve systems of equations using matrices

Be Prepared 4.13

Before you get started, take this readiness quiz.

Solve: $\mathrm{iii}(x+\mathrm{ii})+\mathrm{four}=\mathrm{iv}(\mathrm{ii}x-\mathrm{one})+9.$
If you missed this problem, review Example two.2.

Be Prepared 4.14

Solve: $0.25p+0.25(p+4)=\mathrm{five.20}.$
If you missed this problem, review Example 2.13.

Exist Prepared 4.15

Evaluate when $\mathrm{ten}=\mathrm{-2}$ and $y=3\text{:}\mathrm{ii}{x}^2-\mathrm{10}y+3y^2.$
If you missed this trouble, review Example 1.21.

Write the Augmented Matrix for a System of Equations

Solving a organisation of equations tin can be a boring operation where a elementary mistake can wreak havoc on finding the solution. An alternative method which uses the basic procedures of elimination but with notation that is simpler is bachelor. The method involves using a matrix. A matrix is a rectangular array of numbers bundled in rows and columns.

Matrix

A matrix is a rectangular array of numbers arranged in rows and columns.

A matrix with g rows and n columns has order $m\times n.$ The matrix on the left beneath has 2 rows and 3 columns and then information technology has order $2\times 3.$ We say information technology is a 2 by 3 matrix.

Each number in the matrix is called an chemical element or entry in the matrix.

We will use a matrix to represent a system of linear equations. We write each equation in standard form and the coefficients of the variables and the abiding of each equation becomes a row in the matrix. Each column then would be the coefficients of i of the variables in the system or the constants. A vertical line replaces the equal signs. We call the resulting matrix the augmented matrix for the organization of equations.

Detect the first column is made up of all the coefficients of x, the second column is the all the coefficients of y, and the third column is all the constants.

Instance 4.37

Write each arrangement of linear equations equally an augmented matrix:

ⓐ $\{\begin{array}{c}\mathrm{v}x-\mathrm{iii}y=\mathrm{-1}\hfill \\ y=2x-2\hfill \end{array}$ ⓑ $\{\begin{array}{c}\mathrm{half\; dozen}x-5y+\mathrm{two}z=\mathrm{three}\hfill \\ 2x+y-4z=\mathrm{v}\hfill \\ \mathrm{three}x-\mathrm{iii}y+z=\mathrm{-1}\hfill \end{array}$

Try It iv.73

Write each system of linear equations as an augmented matrix:

ⓐ $\{\begin{array}{c}3\mathrm{ten}+\mathrm{viii}y=\mathrm{-3}\hfill \\ \mathrm{two}\mathrm{ten}=\mathrm{-5}y-3\hfill \end{array}$ ⓑ $\{\begin{array}{c}2x-5y+\mathrm{three}z=8\hfill \\ 3x-y+4z=7\hfill \\ \mathrm{ten}+3y+2z=\mathrm{-3}\hfill \end{array}$

Effort It 4.74

Write each system of linear equations as an augmented matrix:

ⓐ $\{\begin{array}{c}\mathrm{eleven}x=\mathrm{-9}y-5\hfill \\ 7x+\mathrm{five}y=\mathrm{-1}\hfill \end{array}$ ⓑ $\{\begin{array}{c}5\mathrm{ten}-3y+2z=\mathrm{-5}\hfill \\ 2x-y-z=4\hfill \\ 3x-2y+2z=\mathrm{-vii}\hfill \end{array}$

It is important equally we solve systems of equations using matrices to be able to get back and forth between the system and the matrix. The next example asks us to take the information in the matrix and write the system of equations.

Example 4.38

Write the organisation of equations that corresponds to the augmented matrix:

$[\begin{array}{ccccccccc}\hfill 4& & & \hfill \mathrm{-iii}& & & \hfill \mathrm{three}& & \\ \hfill \mathrm{one}& & & \hfill 2& & & \hfill \mathrm{-one}& & \\ \hfill \mathrm{-ii}& & & \hfill \mathrm{-1}& & & \hfill 3& & \end{array}|\begin{array}{ccc}& & \hfill \mathrm{-ane}\\ & & \hfill 2\\ & & \hfill \mathrm{-4}\end{array}].$

Endeavour It 4.75

Write the system of equations that corresponds to the augmented matrix: $\left[\begin{array}{cccccccccc}\mathrm{ane}\hfill & & & \hfill \mathrm{-1}& & & \hfill 2& & & \hfill 3\\ 2\hfill & & & \hfill \mathrm{one}& & & \hfill \mathrm{-2}& & & \hfill \mathrm{one}\\ \mathrm{four}\hfill & & & \hfill \mathrm{-1}& & & \hfill 2& & & \hfill 0\end{array}\right].$

Try Information technology iv.76

Write the organization of equations that corresponds to the augmented matrix: $\left[\begin{array}{cccccccccc}\mathrm{one}\hfill & & & 1\hfill & & & \hfill \mathrm{one}& & & \hfill \mathrm{iv}\\ 2\hfill & & & 3\hfill & & & \hfill \mathrm{-one}& & & \hfill 8\\ 1\hfill & & & 1\hfill & & & \hfill \mathrm{-1}& & & \hfill 3\end{array}\right].$

Use Row Operations on a Matrix

Once a organisation of equations is in its augmented matrix grade, we will perform operations on the rows that will lead us to the solution.

To solve by elimination, it doesn't matter which order we place the equations in the arrangement. Similarly, in the matrix we tin interchange the rows.

When nosotros solve by elimination, we oftentimes multiply one of the equations by a constant. Since each row represents an equation, and we can multiply each side of an equation past a abiding, similarly we can multiply each entry in a row by whatever existent number except 0.

In elimination, we often add together a multiple of one row to another row. In the matrix we can replace a row with its sum with a multiple of another row.

These deportment are called row operations and will assistance us employ the matrix to solve a organization of equations.

Row Operations

In a matrix, the post-obit operations can exist performed on any row and the resulting matrix will be equivalent to the original matrix.

1. Interchange any two rows.
2. Multiply a row by whatsoever real number except 0.
3. Add a nonzero multiple of one row to another row.

Performing these operations is piece of cake to practice but all the arithmetic tin result in a fault. If we use a organization to tape the row functioning in each step, it is much easier to go back and check our work.

We apply upper-case letter letters with subscripts to represent each row. Nosotros then show the operation to the left of the new matrix. To show interchanging a row:

To multiply row 2 by $\mathrm{-3}$ :

To multiply row 2 by $\mathrm{-3}$ and add it to row i:

Case 4.39

Perform the indicated operations on the augmented matrix:

ⓐ Interchange rows 2 and 3.

ⓑ Multiply row 2 past five.

ⓒ Multiply row 3 by $\mathrm{-2}$ and add together to row ane.

$$[\begin{array}{ccccccccc}\mathrm{half\; dozen}\hfill & & & \hfill \mathrm{-5}& & & \hfill 2& & \\ 2\hfill & & & \hfill 1& & & \hfill \mathrm{-iv}& & \\ 3\hfill & & & \hfill \mathrm{-3}& & & \hfill \mathrm{i}& & \end{array}|\begin{array}{ccc}& & \hfill \mathrm{iii}\\ & & \hfill 5\\ & & \hfill \mathrm{-1}\end{array}]$$

Try It four.77

Perform the indicated operations sequentially on the augmented matrix:

ⓐ Interchange rows i and 3.

ⓑ Multiply row iii past three.

ⓒ Multiply row 3 by ii and add to row 2.

$[\begin{array}{ccccccccc}\hfill 5& & & \hfill \mathrm{-2}& & & \hfill \mathrm{-two}& & \\ \hfill 4& & & \hfill \mathrm{-1}& & & \hfill \mathrm{-4}& & \\ \hfill \mathrm{-two}& & & \hfill 3& & & \hfill 0& & \end{array}|\begin{array}{ccc}& & \hfill \mathrm{-2}\\ & & \hfill 4\\ & & \hfill \mathrm{-1}\end{array}]$

Attempt It 4.78

Perform the indicated operations on the augmented matrix:

ⓐ Interchange rows 1 and 2,

ⓑ Multiply row 1 by 2,

ⓒ Multiply row 2 by 3 and add to row i.

$[\begin{array}{ccccccccc}\mathrm{ii}\hfill & & & \hfill \mathrm{-3}& & & \hfill \mathrm{-2}& & \\ \mathrm{four}\hfill & & & \hfill \mathrm{i}& & & \hfill \mathrm{-3}& & \\ 5\hfill & & & \hfill 0& & & \hfill \mathrm{iv}& & \end{array}|\begin{array}{ccc}& & \hfill \mathrm{-four}\\ & & \hfill 2\\ & & \hfill \mathrm{-1}\end{array}]$

Now that we have skillful the row operations, nosotros volition look at an augmented matrix and figure out what operation we volition employ to achieve a goal. This is exactly what nosotros did when we did elimination. Nosotros decided what number to multiply a row by in order that a variable would be eliminated when we added the rows together.

Given this system, what would you do to eliminate x?

This next case essentially does the same affair, but to the matrix.

Instance 4.40

Perform the needed row operation that will get the get-go entry in row ii to exist nil in the augmented matrix: $[\begin{array}{cccccc}1\hfill & & & \hfill \mathrm{-1}& & \\ \mathrm{four}\hfill & & & \hfill \mathrm{-8}& & \end{array}|\begin{array}{ccc}& & \hfill 2\\ & & \hfill 0\end{array}].$

Effort It 4.79

Perform the needed row operation that volition get the first entry in row ii to be naught in the augmented matrix: $[\begin{array}{cccccc}1\hfill & & & \hfill \mathrm{-1}& & \\ 3\hfill & & & \hfill \mathrm{-6}& & \end{array}|\begin{array}{ccc}& & \hfill 2\\ & & \hfill 2\end{array}].$

Endeavor It four.80

Perform the needed row performance that will become the first entry in row two to be zero in the augmented matrix: $[\begin{array}{cccccc}\hfill 1& & & \hfill \mathrm{-1}& & \\ \hfill \mathrm{-2}& & & \hfill \mathrm{-3}& & \end{array}|\begin{array}{ccc}& & \hfill 3\\ & & \hfill 2\end{array}].$

Solve Systems of Equations Using Matrices

To solve a organization of equations using matrices, we transform the augmented matrix into a matrix in row-echelon form using row operations. For a consistent and independent system of equations, its augmented matrix is in row-echelon grade when to the left of the vertical line, each entry on the diagonal is a 1 and all entries below the diagonal are zeros.

Row-Echelon Form

For a consistent and independent system of equations, its augmented matrix is in row-echelon form when to the left of the vertical line, each entry on the diagonal is a 1 and all entries below the diagonal are zeros.

Once we get the augmented matrix into row-echelon form, we tin write the equivalent system of equations and read the value of at least 1 variable. We then substitute this value in some other equation to keep to solve for the other variables. This process is illustrated in the adjacent instance.

Case four.41

How to Solve a System of Equations Using a Matrix

Solve the arrangement of equations using a matrix: $\{\begin{array}{c}3\mathrm{ten}+4y=5\hfill \\ \mathrm{10}+2y=1\hfill \end{array}.$

Endeavor It 4.81

Solve the system of equations using a matrix: $\{\begin{array}{c}2x+y=7\hfill \\ x-2y=6\hfill \end{array}.$

Effort It 4.82

Solve the organization of equations using a matrix: $\{\begin{array}{c}2x+y=\mathrm{-4}\hfill \\ x-y=\mathrm{-two}\hfill \end{array}.$

The steps are summarized here.

How To

Solve a system of equations using matrices.

1. Step ane. Write the augmented matrix for the organisation of equations.
2. Footstep ii. Using row operations get the entry in row one, cavalcade ane to be 1.
3. Stride iii. Using row operations, get zeros in column one below the 1.
4. Stride 4. Using row operations, get the entry in row 2, column 2 to be one.
5. Step 5. Continue the process until the matrix is in row-echelon class.
6. Step half dozen. Write the corresponding system of equations.
7. Step 7. Use exchange to detect the remaining variables.
8. Footstep 8. Write the solution as an ordered pair or triple.
9. Stride 9. Check that the solution makes the original equations true.

Here is a visual to prove the club for getting the 1'southward and 0's in the proper position for row-echelon class.

We apply the same procedure when the arrangement of equations has iii equations.

Instance 4.42

Solve the system of equations using a matrix: $\{\begin{array}{c}3x+\mathrm{viii}y+2z=\mathrm{-5}\hfill \\ 2x+\mathrm{five}y-\mathrm{three}z=0\hfill \\ x+2y-\mathrm{two}z=\mathrm{-1}\hfill \end{array}.$

Try Information technology 4.83

Solve the organisation of equations using a matrix: $\{\begin{array}{c}\mathrm{two}x-\mathrm{five}y+\mathrm{three}z=\mathrm{eight}\hfill \\ 3\mathrm{10}-y+4z=\mathrm{vii}\hfill \\ \mathrm{10}+3y+\mathrm{two}z=\mathrm{-three}\hfill \end{array}.$

Effort It 4.84

Solve the system of equations using a matrix: $\{\begin{array}{c}\hfill \mathrm{-three}x+y+z=\mathrm{-4}\\ \hfill \text{−}x+\mathrm{two}y-2z=1\\ 2x-y-z=\mathrm{-i}\hfill \end{array}.$

And so far our work with matrices has only been with systems that are consistent and contained, which means they have exactly one solution. Permit's at present expect at what happens when we use a matrix for a dependent or inconsistent system.

Example 4.43

Solve the system of equations using a matrix: $\{\begin{array}{c}x+y+\mathrm{three}z=0\hfill \\ x+3y+5z=0\hfill \\ 2\mathrm{ten}+4z=1\hfill \end{array}.$

Try It iv.85

Solve the system of equations using a matrix: $\{\begin{array}{c}\mathrm{ten}-2y+2z=1\hfill \\ \hfill \mathrm{-2}x+y-z=2\\ \mathrm{10}-y+z=\mathrm{five}\hfill \end{array}.$

Effort It four.86

Solve the system of equations using a matrix: $\{\begin{array}{c}3x+4y-3z=\mathrm{-ii}\hfill \\ 2x+3y-z=\mathrm{-12}\hfill \\ x+y-2z=\mathrm{half-dozen}\hfill \end{array}.$

The terminal system was inconsistent and so had no solutions. The next example is dependent and has infinitely many solutions.

Case 4.44

Solve the system of equations using a matrix: $\{\begin{array}{c}\mathrm{ten}-2y+3z=1\hfill \\ \mathrm{10}+y-\mathrm{three}z=7\hfill \\ 3\mathrm{10}-\mathrm{iv}y+\mathrm{v}z=7\hfill \end{array}.$

Endeavour It 4.87

Solve the organisation of equations using a matrix: $\{\begin{array}{c}x+y-z=0\hfill \\ \mathrm{ii}\mathrm{10}+\mathrm{iv}y-2z=6\hfill \\ 3\mathrm{ten}+6y-\mathrm{three}z=9\hfill \end{array}.$

Attempt It 4.88

Solve the system of equations using a matrix: $\{\begin{array}{c}x-y-z=1\hfill \\ \hfill \text{−}x+2y-\mathrm{iii}z=\mathrm{-4}\\ 3\mathrm{ten}-\mathrm{ii}y-7z=0\hfill \end{array}.$

Section 4.5 Exercises

Practice Makes Perfect

Write the Augmented Matrix for a System of Equations

In the following exercises, write each organization of linear equations as an augmented matrix.

196 .

ⓐ $\{\begin{array}{c}3\mathrm{10}-y=\mathrm{-1}\hfill \\ 2y=\mathrm{ii}x+5\hfill \end{array}$
ⓑ $\{\begin{array}{c}\mathrm{iv}x+3y=\mathrm{-2}\hfill \\ \mathrm{10}-2y-\mathrm{iii}z=7\hfill \\ 2\mathrm{ten}-y+2z=\mathrm{-half\; dozen}\hfill \end{array}$

197.

ⓐ $\{\begin{array}{c}\mathrm{ii}x+\mathrm{four}y=\mathrm{-5}\hfill \\ 3x-2y=2\hfill \end{array}$
ⓑ $\{\begin{array}{c}\mathrm{iii}\mathrm{ten}-2y-z=\mathrm{-two}\hfill \\ \mathrm{-2}x+y=\mathrm{v}\hfill \\ \mathrm{five}x+4y+z=\mathrm{-ane}\hfill \end{array}$

198 .

ⓐ $\{\begin{array}{c}\mathrm{three}x-y=\mathrm{-4}\hfill \\ \mathrm{ii}x=y+2\hfill \end{array}$
ⓑ $\{\begin{array}{c}\mathrm{10}-3y-\mathrm{iv}z=\mathrm{-2}\hfill \\ 4x+\mathrm{ii}y+\mathrm{ii}z=5\hfill \\ 2x-5y+7z=\mathrm{-eight}\hfill \end{array}$

199.

ⓐ $\{\begin{array}{c}\mathrm{two}x-\mathrm{v}y=\mathrm{-3}\hfill \\ 4x=3y-1\hfill \end{array}$
ⓑ $\{\begin{array}{c}4x+3y-2z=\mathrm{-3}\hfill \\ \hfill \mathrm{-2}\mathrm{10}+y-3z=\mathrm{four}\\ \hfill \text{−}x-4y+5z=\mathrm{-2}\end{array}$

Write the arrangement of equations that corresponds to the augmented matrix.

200 .

$[\begin{array}{cccc}\mathrm{ii}\hfill & & & \hfill \mathrm{-1}\\ \mathrm{one}\hfill & & & \hfill \mathrm{-three}\end{array}|\begin{array}{c}\hfill 4\\ \hfill 2\end{array}]$

201.

$[\begin{array}{cccc}\mathrm{ii}\hfill & & & \hfill \mathrm{-4}\\ 3\hfill & & & \hfill \mathrm{-three}\end{array}|\begin{array}{c}\hfill \mathrm{-two}\\ \hfill \mathrm{-1}\end{array}]$

202 .

$[\begin{array}{ccccccc}\mathrm{i}\hfill & & & \hfill 0& & & \hfill \mathrm{-3}\\ 1\hfill & & & \hfill \mathrm{-two}& & & \hfill 0\\ 0\hfill & & & \hfill \mathrm{-1}& & & \hfill \mathrm{two}\end{array}|\begin{array}{c}\hfill \mathrm{-1}\\ \hfill \mathrm{-2}\\ \hfill 3\end{array}]$

203.

$[\begin{array}{ccccccc}2\hfill & & & \hfill \mathrm{-2}& & & \hfill 0\\ 0\hfill & & & \hfill \mathrm{two}& & & \hfill \mathrm{-1}\\ 3\hfill & & & \hfill 0& & & \hfill \mathrm{-1}\end{array}|\begin{array}{c}\hfill \mathrm{-i}\\ \hfill 2\\ \hfill \mathrm{-2}\end{array}]$

Use Row Operations on a Matrix

In the following exercises, perform the indicated operations on the augmented matrices.

204 .

$[\begin{array}{cccc}6\hfill & & & \hfill \mathrm{-iv}\\ 3\hfill & & & \hfill \mathrm{-2}\end{array}|\begin{array}{c}\hfill 3\\ \hfill 1\end{array}]$

ⓐ Interchange rows 1 and 2

ⓑ Multiply row 2 by iii

ⓒ Multiply row 2 by $\mathrm{-2}$ and add row one to information technology.

205.

$[\begin{array}{cccc}\mathrm{four}\hfill & & & \hfill \mathrm{-6}\\ 3\hfill & & & \hfill 2\end{array}|\begin{array}{c}\hfill \mathrm{-3}\\ \hfill 1\end{array}]$

ⓐ Interchange rows ane and 2

ⓑ Multiply row 1 by four

ⓒ Multiply row 2 by three and add together row i to it.

206 .

$[\begin{array}{ccccccc}\mathrm{four}\hfill & & & \hfill \mathrm{-12}& & & \hfill \mathrm{-viii}\\ 4\hfill & & & \hfill \mathrm{-2}& & & \hfill \mathrm{-3}\\ \mathrm{-6}\hfill & & & \hfill 2& & & \hfill \mathrm{-1}\end{array}|\begin{array}{c}\hfill \mathrm{sixteen}\\ \hfill \mathrm{-i}\\ \hfill \mathrm{-one}\end{array}]$

ⓐ Interchange rows 2 and 3

ⓑ Multiply row ane by 4

ⓒ Multiply row 2 by $\mathrm{-two}$ and add to row 3.

207.

$[\begin{array}{ccccccc}6\hfill & & & \hfill \mathrm{-v}& & & \hfill 2\\ \mathrm{ii}\hfill & & & \hfill 1& & & \hfill \mathrm{-4}\\ 3\hfill & & & \hfill \mathrm{-3}& & & \hfill 1\end{array}|\begin{array}{c}\hfill \mathrm{iii}\\ \hfill 5\\ \hfill \mathrm{-1}\end{array}]$

ⓐ Interchange rows 2 and three

ⓑ Multiply row 2 by v

ⓒ Multiply row iii by $\mathrm{-2}$ and add together to row 1.

208 .

Perform the needed row operation that volition get the first entry in row 2 to be zero in the augmented matrix: $[\begin{array}{cccc}\hfill 1& & & \hfill \mathrm{two}\\ \hfill \mathrm{-3}& & & \hfill \mathrm{-4}\end{array}|\begin{array}{c}\hfill 5\\ \hfill \mathrm{-1}\end{array}].$

209.

Perform the needed row operations that volition get the first entry in both row ii and row three to exist zero in the augmented matrix: $[\begin{array}{ccccccc}1\hfill & & & \hfill \mathrm{-2}& & & \hfill \mathrm{iii}\\ 3\hfill & & & \hfill \mathrm{-i}& & & \hfill \mathrm{-2}\\ 2\hfill & & & \hfill \mathrm{-3}& & & \hfill \mathrm{-4}\end{array}|\begin{array}{c}\hfill \mathrm{-4}\\ \hfill 5\\ \hfill \mathrm{-1}\end{array}].$

Solve Systems of Equations Using Matrices

In the following exercises, solve each system of equations using a matrix.

210 .

$\{\begin{array}{c}2x+y=\mathrm{two}\hfill \\ x-y=\mathrm{-two}\hfill \end{array}$

211.

$\{\begin{array}{c}\mathrm{iii}x+y=2\hfill \\ \mathrm{10}-y=2\hfill \end{array}$

212 .

$\{\begin{array}{c}\hfill \text{−}x+2y=\mathrm{-2}\\ \mathrm{10}+y=\mathrm{-iv}\hfill \end{array}$

213.

$\{\begin{array}{c}\hfill \mathrm{-2}\mathrm{10}+3y=3\\ x+3y=12\hfill \end{array}$

In the following exercises, solve each arrangement of equations using a matrix.

214 .

$\{\begin{array}{c}2\mathrm{ten}-\mathrm{three}y+z=\mathrm{xix}\hfill \\ \hfill \mathrm{-3}x+y-2z=\mathrm{-15}\\ \mathrm{10}+y+z=0\hfill \end{array}$

215.

$\{\begin{array}{c}2x-y+3z=\mathrm{-3}\hfill \\ \hfill \text{−}x+\mathrm{ii}y-z=\mathrm{x}\\ x+y+z=5\hfill \end{array}$

216 .

$\{\begin{array}{c}\mathrm{ii}x-6y+z=\mathrm{iii}\hfill \\ 3x+2y-\mathrm{three}z=2\hfill \\ 2\mathrm{10}+\mathrm{iii}y-\mathrm{two}z=\mathrm{iii}\hfill \end{array}$

217.

$\{\begin{array}{c}\mathrm{iv}\mathrm{10}-3y+z=\mathrm{seven}\hfill \\ 2x-5y-4z=\mathrm{iii}\hfill \\ \mathrm{iii}\mathrm{ten}-\mathrm{two}y-\mathrm{two}z=\mathrm{-7}\hfill \end{array}$

218 .

$\{\begin{array}{c}x+2z=0\hfill \\ 4y+\mathrm{three}z=\mathrm{-2}\hfill \\ 2x-\mathrm{v}y=\mathrm{iii}\hfill \end{array}$

219.

$\{\begin{array}{c}2\mathrm{ten}+5y=4\hfill \\ 3y-z=\mathrm{three}\hfill \\ 4x+\mathrm{three}z=\mathrm{-three}\hfill \end{array}$

220 .

$\{\begin{array}{c}2y+3z=\mathrm{-1}\hfill \\ \mathrm{v}\mathrm{10}+3y=\mathrm{-half\; dozen}\hfill \\ \mathrm{seven}\mathrm{10}+z=1\hfill \end{array}$

221.

$\{\begin{array}{c}\mathrm{three}x-z=\mathrm{-3}\hfill \\ \mathrm{five}y+\mathrm{two}z=\mathrm{-half-dozen}\hfill \\ \mathrm{four}x+3y=\mathrm{-8}\hfill \end{array}$

222 .

$\{\begin{array}{c}\mathrm{ii}\mathrm{ten}+3y+z=12\hfill \\ x+y+z=\mathrm{nine}\hfill \\ 3\mathrm{ten}+4y+2z=20\hfill \end{array}$

223.

$\{\begin{array}{c}x+\mathrm{two}y+\mathrm{six}z=\mathrm{v}\hfill \\ \hfill \text{−}x+y-\mathrm{ii}z=3\\ \mathrm{10}-\mathrm{iv}y-2z=1\hfill \end{array}$

224 .

$\{\begin{array}{c}x+2y-\mathrm{three}z=\mathrm{-1}\hfill \\ x-3y+z=\mathrm{ane}\hfill \\ \mathrm{ii}\mathrm{ten}-y-\mathrm{two}z=2\hfill \end{array}$

225.

$\{\begin{array}{c}\mathrm{four}x-3y+2z=0\hfill \\ \hfill \mathrm{-2}\mathrm{ten}+3y-7z=1\\ 2\mathrm{10}-\mathrm{ii}y+3z=6\hfill \end{array}$

226 .

$\{\begin{array}{c}\mathrm{10}-y+2z=\mathrm{-4}\hfill \\ 2\mathrm{10}+y+3z=2\hfill \\ \hfill \mathrm{-iii}x+3y-6z=12\end{array}$

227.

$\{\begin{array}{c}\hfill \text{−}x-3y+\mathrm{two}z=14\\ \hfill \text{−}\mathrm{10}+2y-\mathrm{three}z=\mathrm{-4}\\ \mathrm{iii}\mathrm{10}+y-2z=6\hfill \end{array}$

228 .

$\{\begin{array}{c}x+y-3z=\mathrm{-ane}\hfill \\ y-z=0\hfill \\ \hfill \text{−}x+\mathrm{ii}y=1\end{array}$

229.

$\{\begin{array}{c}x+2y+z=4\hfill \\ x+y-2z=3\hfill \\ \hfill \mathrm{-2}x-3y+z=\mathrm{-7}\end{array}$

Writing Exercises

230 .

Solve the system of equations $\{\begin{array}{c}x+y=10\hfill \\ x-y=\mathrm{six}\hfill \end{array}$ ⓐ by graphing and ⓑ past substitution. ⓒ Which method do you prefer? Why?

231.

Solve the system of equations $\{\begin{array}{c}3x+y=12\hfill \\ \mathrm{ten}=y-8\hfill \end{array}$ by substitution and explain all your steps in words.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After looking at the checklist, do you think you are well-prepared for the next department? Why or why non?

Matrix With Infinitely Many Solutions,

Source: https://openstax.org/books/intermediate-algebra-2e/pages/4-5-solve-systems-of-equations-using-matrices

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