Perform the row operation on (row ) in order to convert some elements in the row to . Show
Replace (row ) with the row operation in order to convert some elements in the row to the desired value . Replace (row ) with the actual values of the elements for the row operation . Perform the row operation on (row ) in order to convert some elements in the row to . Replace (row ) with the row operation in order to convert some elements in the row to the desired value . Replace (row ) with the actual values of the elements for the row operation . Perform the row operation on (row ) in order to convert some elements in the row to . Replace (row ) with the row operation in order to convert some elements in the row to the desired value . Replace (row ) with the actual values of the elements for the row operation . Perform the row operation on (row ) in order to convert some elements in the row to . Replace (row ) with the row operation in order to convert some elements in the row to the desired value . Replace (row ) with the actual values of the elements for the row operation .
Learning ObjectivesBy the end of this section, you will be able to:
Be Prepared 4.13Before you get started, take this readiness quiz. Solve: 3(x+2)+4=4(2x−1)+9.
3(x+2)+4=4(2x−1)+9.
Be Prepared 4.14Solve: 0.25p+0.25(p+4)=5.20.0.25p+0.25(p+4)=
5.20.
Be Prepared 4.15Evaluate when x=−2x=−2 and y=3:2x2
−xy+3y2.y=3:2x2−xy+3y2. Write the Augmented Matrix for a System of EquationsSolving a system of equations can be a tedious operation where a simple 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 available. The method involves using a matrix. A matrix is a rectangular array of numbers arranged in rows and columns.
MatrixA matrix is a rectangular array of numbers arranged in rows and columns. A matrix with m rows and n columns has order m×n.m ×n. The matrix on the left below has 2 rows and 3 columns and so it has order 2×3.2×3. We say it is a 2 by 3 matrix.
Each number in the matrix is called an 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 constant of each equation becomes a row in the matrix. Each column then would be the coefficients of one 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 system of equations. Notice 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.
Example 4.37Write each system of linear equations as an augmented matrix: ⓐ {5x−3y=−1y=2x −2{5x−3y=−1y=2x−2 ⓑ { 6x−5y+2z=32x+y−4z=53x−3y+z=−1{ 6x−5y+2z=32x+y−4z=53x−3y+z=−1
Try It 4.73Write each system of linear equations as an augmented matrix: ⓐ {3x+8y=−32x=−5y−3 {3x+8y=−32x=−5y−3 ⓑ {2x−5y+3z=83x−y+4z=7x+3y+2z=−3{ 2x−5y+3z=83x−y+4z=7x+3y+2z=−3
Try It 4.74Write each system of linear equations as an augmented matrix: ⓐ {11x=−9y−57x+5y=−1 {11x=−9y−57x+5y=−1 ⓑ {5x−3y+2z=−52x−y−z=43x−2y+2z=−7{ 5x−3y+2z=−52x−y−z=43x−2y+2z=−7 It is important as we solve systems of equations using matrices to be able to go 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 4.38Write the system of equations that corresponds to the augmented matrix: [4−33 12−1−2−13| −12−4].[4−3312−1−2−13| −12−4].
Try It 4.75Write the system of equations that corresponds to the augmented matrix: [1−12321−214−120] .[1−12321−2 14−120].
Try It 4.76Write the system of equations that corresponds to the augmented matrix: [111423−1811−13]. [111423−1 811−13]. Use Row Operations on a MatrixOnce a system of equations is in its augmented matrix form, 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 system. Similarly, in the matrix we can interchange the rows. When we solve by elimination, we often multiply one of the equations by a constant. Since each row represents an equation, and we can multiply each side of an equation by a constant, similarly we can multiply each entry in a row by any real number except 0. In elimination, we often add 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 actions are called row operations and will help us use the matrix to solve a system of equations.
Row OperationsIn a matrix, the following operations can be performed on any row and the resulting matrix will be equivalent to the original matrix.
Performing these operations is easy to do but all the arithmetic can result in a mistake. If we use a system to record the row operation in each step, it is much easier to go back and check our work. We use capital letters with subscripts to represent each row. We then show the operation to the left of the new matrix. To show interchanging a row: To multiply row 2 by −3−3 : To multiply row 2 by −3−3 and add it to row 1:
Example 4.39Perform the indicated operations on the augmented matrix: ⓐ Interchange rows 2 and 3. ⓑ Multiply row 2 by 5. ⓒ Multiply row 3 by −2−2 and add to row 1. [6−5221 −43−31|35 −1][6−5221 −43−31|35 −1]
Try It 4.77Perform the indicated operations sequentially on the augmented matrix: ⓐ Interchange rows 1 and 3. ⓑ Multiply row 3 by 3. ⓒ Multiply row 3 by 2 and add to row 2. [ 5−2−24−1−4−23 0|−24−1][ 5−2−24−1−4−2 30|−24−1]
Try It 4.78Perform 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 1. [2−3−241−350 4|−42−1][ 2−3−241−350 4|−42−1] Now that we have practiced the row operations, we will look at an augmented matrix and figure out what operation we will use to reach a goal. This is exactly what we did when we did elimination. We 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 example essentially does the same thing, but to the matrix.
Example 4.40Perform the needed row operation that will get the first entry in row 2 to be zero in the augmented matrix: [1−14−8|20].[1 −14−8|20].
Try It 4.79Perform the needed row operation that will get the first entry in row 2 to be zero in the augmented matrix: [1−13−6|22].[1−1 3−6|22].
Try It 4.80Perform the needed row operation that will get the first entry in row 2 to be zero in the augmented matrix: [1−1−2−3|32].[1−1 −2−3|32]. Solve Systems of Equations Using MatricesTo solve a system 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 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.
Row-Echelon FormFor 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 can write the equivalent system of equations and read the value of at least one variable. We then substitute this value in another equation to continue to solve for the other variables. This process is illustrated in the next example.
Example 4.41How to Solve a System of Equations Using a MatrixSolve the system of equations using a matrix: {3x+4y=5x +2y=1.{3x+4y=5x+2y=1.
Try It 4.81Solve the system of equations using a matrix: {2x+y=7x−2y=6. {2x+y=7x−2y=6.
Try It 4.82Solve the system of equations using a matrix: {2x+y=−4x−y=−2.{ 2x+y=−4x−y=−2. The steps are summarized here.
How ToSolve a system of equations using matrices.
Here is a visual to show the order for getting the 1’s and 0’s in the proper position for row-echelon form. We use the same procedure when the system of equations has three equations.
Example 4.42Solve the system of equations using a matrix: {3x+8y+2z=−52x+5y−3 z=0x+2y−2z=−1.{3x+8y+2z=−52x+5 y−3z=0x+2y−2z=−1.
Try It 4.83Solve the system of equations using a matrix: {2x−5y+3z=83x−y+4z=7x+3y+2z=−3.{2x−5y+3z=83x−y+4z=7 x+3y+2z=−3.
Try It 4.84Solve the system of equations using a matrix: {−3x+y+z=−4−x+2y−2z=12x−y−z=−1.{−3x+y+z=−4−x+2y−2z=1 2x−y−z=−1. So far our work with matrices has only been with systems that are consistent and independent, which means they have exactly one solution. Let’s now look at what happens when we use a matrix for a dependent or inconsistent system.
Example 4.43Solve the system of equations using a matrix: {x+y+3z=0x+3y+5z=0 2x+4z=1.{x+y+3z=0x+3y+5z=02x+4z=1.
Try It 4.85Solve the system of equations using a matrix: {x−2y+2z=1−2x+y−z=2x −y+z=5.{x−2y+2z=1−2x+y−z=2x−y +z=5.
Try It 4.86Solve the system of equations using a matrix: {3x+4y−3z=−22x+3y−z=−12x+y−2z=6.{3x+4y−3z=−22x+3y−z=−12 x+y−2z=6. The last system was inconsistent and so had no solutions. The next example is dependent and has infinitely many solutions.
Example 4.44Solve the system of equations using a matrix: {x−2y+3z=1x+y−3z=7 3x−4y+5z=7.{x−2y+3z=1x+y−3z=7 3x−4y+5z=7.
Try It 4.87Solve the system of equations using a matrix: {x+y−z=02x+4y−2z=63 x+6y−3z=9.{x+y−z=02x+4y−2z=63 x+6y−3z=9.
Try It 4.88Solve the system of equations using a matrix: {x−y−z=1−x+2y−3z=−4 3x−2y−7z=0.{x−y−z=1−x+2y−3z=−43x−2y−7z=0. Section 4.5 ExercisesPractice Makes PerfectWrite the Augmented Matrix for a System of Equations In the following exercises, write each system of linear equations as an augmented matrix. 196.
197.
198.
199.
Write the system of equations that corresponds to the augmented matrix. 200. [2−11−3|42][2−11 −3|42] 201. [2−43−3|−2−1][2 −43−3|−2−1] 202. [10−31−2 00−12|−1−23 ][10−31−20 0−12|−1−23] 203. [2 −2002−130−1 |−12−2][2−2 002−130−1| −12−2] Use Row Operations on a Matrix In the following exercises, perform the indicated operations on the augmented matrices. 204. [6−43−2|31][6−43−2 |31] ⓐ Interchange rows 1 and 2 ⓑ Multiply row 2 by 3 ⓒ Multiply row 2 by −2−2 and add row 1 to it. 205. [4−632|−31][4 −632|−31] ⓐ Interchange rows 1 and 2 ⓑ Multiply row 1 by 4 ⓒ Multiply row 2 by 3 and add row 1 to it. 206. [4 −12−84−2−3−62−1|16−1−1][4−12 −84−2−3−62−1| 16−1−1] ⓐ Interchange rows 2 and 3 ⓑ Multiply row 1 by 4 ⓒ Multiply row 2 by −2−2 and add to row 3. 207. [6−5221−43−31|35−1][6−5221−43−31|35−1] ⓐ Interchange rows 2 and 3 ⓑ Multiply row 2 by 5 ⓒ Multiply row 3 by −2−2 and add to row 1. 208. Perform the needed row operation that will get the first entry in row 2 to be zero in the augmented matrix: [12−3−4| 5−1].[12−3−4 |5−1]. 209. Perform the needed row operations that will get the first entry in both row 2 and row 3 to be zero in the augmented matrix: [1−233−1−22−3−4|−45−1].[1 −233−1−22−3−4 |−45−1]. Solve Systems of Equations Using Matrices In the following exercises, solve each system of equations using a matrix. 210. {2x+y=2x−y=−2{2x+y=2x−y=−2 211. {3x+y=2x− y=2{3x+y=2x−y=2 212. {−x+2y=−2x+y=−4{−x+2y =−2x+y=−4 213. {−2x+3y=3x+3y=12{−2x+3y =3x+3y=12 In the following exercises, solve each system of equations using a matrix. 214. { 2x−3y+z=19−3x+y−2z=−15x+y+z=0{2x−3y+z=19−3x+y−2z=−15x+y+z=0 215. {2x−y+3z=−3−x+2y−z=10x+y+z=5{2x−y+3z=−3−x+2y−z=10x+y+z=5 216. { 2x−6y+z=33x+2y−3z=22x+3y−2z=3 {2x−6y+z=33x+2y−3z=22x+3y−2z=3 217. {4x−3y+ z=72x−5y−4z=33x−2y−2z=−7{4x−3 y+z=72x−5y−4z=33x−2y−2z=−7 218. {x+2z=04y+3z=−22x−5y=3{x+2z=04y+3z=−22x−5y=3 219. {2x+5y=43y−z=34x+3z=−3{2x+5y=43y−z=3 4x+3z=−3 220. {2y+3z=−1 5x+3y=−67x+z=1{2y+3z=−15x+ 3y=−67x+z=1 221. {3x−z=−35y +2z=−64x+3y=−8{3x−z=−35y+2z=−64x+3y=−8 222. {2x+3y+z=12 x+y+z=93x+4y+2z=20{2x+3y+z=12 x+y+z=93x+4y+2z=20 223. {x+2y+6z=5−x+y−2z=3x−4y−2z=1{x+2y+6z=5 −x+y−2z=3x−4y−2z=1 224. {x+2y−3z=−1x−3y+z=12x−y−2z=2 {x+2y−3z=−1x−3y+z=12x−y−2z=2 225. {4x−3y+2 z=0−2x+3y−7z=12x−2y+3z=6{4x−3 y+2z=0−2x+3y−7z=12x−2y+3z=6 226. {x−y+2z=−42x+y+3z=2−3x+3y−6z=12 {x−y+2z=−42x+y+3z=2−3x+3y−6z=12 227. { −x−3y+2z=14−x+2y−3z=−43x+y−2z=6 {−x−3y+2z=14−x+2y−3z=−43x+y−2z=6 228. {x+y−3z=−1y−z=0−x +2y=1{x+y−3z=−1y−z=0−x+2y=1 229. {x+2y+z=4x+y−2z=3−2x−3y+z=−7{ x+2y+z=4x+y−2z=3−2x−3y+z=−7 Writing Exercises230. Solve the system of equations {x+y=10x−y=6{x+y=10x−y=6 ⓐ by graphing and ⓑ by substitution. ⓒ Which method do you prefer? Why? 231. Solve the system of equations {3x+y=12x=y−8{3x+y=12x=y−8 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 section? Why or why not? Can any system of linear equations be written as an augmented matrix?A system of equations can be represented by an augmented matrix. In an augmented matrix, each row represents one equation in the system and each column represents a variable or the constant terms. In this way, we can see that augmented matrices are a shorthand way of writing systems of equations.
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