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Common Core Standards: Math

High School: Algebra

Reasoning with Equations and Inequalities HSA-REI.C.8

8. Represent a system of linear equations as a single matrix equation in a vector variable.

Students should understand that a system of equations can be expressed as a matrix of coefficients multiplied by a vector matrix of variables. Sometimes, it's even better to do it that way.

Your students will probably curse you for stressing matrices time and time again, but just reassure them that it's for their own good. In order to work with matrices, your students should already understand what they are and how to perform various functions with them. The same goes for linear equations and systems of linear equations.

Students should know that a matrix equation takes the form AX = B, where A represents the coefficient of our variables, X represents our variables, and B represents the output to our equations.

In order to turn linear equations into matrices, we have to rearrange them all into the same format. Among the preferred formats are ax + by + cz = d. The linear equations y = 2x + 5 and y = 3x – 2 become -2x + y = 5 and -3x + y = -2. Painless, no? 

Then we take each term in the equations and split them up into their proper matrix. Taking the equations from above, we'll have the A matrix (the one with all the coefficients) equaling 

Since our only variables are x and y (and we put the coefficients into the A matrix in that order), our X matrix becomes:

The final matrix, B, is what's on the other side of our equal signs. That means it turns into:

In matrix form, it looks like this:

This standard only indicates that students should know how to represent systems of linear equations in matrix equation form, but not why doing so is useful (aside from organizational purposes). That is covered in the next standard.

Drills

  1. What is the correct matrix form of the equations 3x + 15y – 37z = 45, 48x + 19z + 66y = 49, and 23x + 66y + 13 +19z = 66?

    Correct Answer:

    Answer Explanation:

    You first have to be careful and actually put the equations into proper form first because some of them have the variable switched around and the last equation needs 13 subtracted from both sides.


  2. What are the corresponding linear equations to the following matrix?

    Correct Answer:

    16x + 88y = 2; 19x + y + 3z = 19; 21x + 3z = 9

    Answer Explanation:

    This problem is simply the "art" of taking the numbers in matrix equation form and assigning the proper coefficients to the variables and the output. The only mistake that you may have made is assigning a 0 value in the matrix a variable when there shouldn't be one. But Shmoopers like you are smart, so it's unlikely that you fell for that.


  3. Put the equations y = 3x + z, z = 77x – 2y + 34, and x = 55x + 31y into proper matrix equation form.

    Correct Answer:

    Answer Explanation:

    The first order of business is to make sure all the equations are in the same format. That means 0 = 3xy + z, -34 = 77x – 2yz, and 0 = 54x + 31y. Then, all it takes is transferring them over into matrix form. Also, don't get confused by (D) because the variables are in the incorrect order. Variables are always changing things up on you.


  4. What is the correct matrix form for the equations x + y = 3 and 2x + 3y = 4?

    Correct Answer:

    Answer Explanation:

    In the first equation, the coefficient for both x and y is one. The rest is simply application of the matrix equation AX = B.


  5. If the equations 34z + 24b = 13 and 24z + 34b = 13 were put into matrix form, what would the A matrix look like?

    Correct Answer:

    Answer Explanation:

    All other answers do not put these equations into the correct matrix format. This is a simple a matter of putting the coefficients into the right places.


  6. What is the proper matrix form of the equations y = 2x + 3 and y = 3x + 5?

    Correct Answer:

    Answer Explanation:

    If we organize the equations into the proper format, we will have y – 2x = 3 and y – 3x = 5. If we put this into matrix form we will see that (A) is the only correct answer, since (B) switches the variables, (D) switches the entire product matrix, and (C) has the incorrect numbers and order of numbers.


  7. The correct form of a matrix equation is:

    Correct Answer:

    AX = B

    Answer Explanation:

    The only answer that makes sense is (C), since (A) is the slope-intercept form of a linear equation, (B) is the equation for a circle, and (D) is the standard form of a quadratic equation.


  8. A matrix equation is used for:

    Correct Answer:

    Both (B) and (C)

    Answer Explanation:

    We use it to solve for the variable in linear equations. So what if we don't know exactly how yet? We can translate linear equations to matrices and back. Isn't that enough?


  9. In the matrix equation AX = B, X represents the:

    Correct Answer:

    Variable

    Answer Explanation:

    In our matrix equation, matrix A represents the coefficients in the linear equations and matrix B represents the output.


  10. What would the B matrix for the equations x + 2y = 3 and x + 2y = 4 be?

    Correct Answer:

    Answer Explanation:

    The B matrix in our matrix equation is the solution of the two linear equations. Since 3 and 4 are our values for the two equations, we list them out in matrix B.