Representing Linear Systems with Matrices - Algebra 2

1. What matrix equation represents $y-3x=2$ and $5x+2y=1$ using variable order $x,y$?

   $\begin{bmatrix}-3&1\\5&2\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}2\\1\end{bmatrix}$. Reorder $y-3x=2$ to $-3x+y=2$ for standard variable order.

2. What is the size of $A$ for a system of $3$ equations in $2$ variables written as $A\vec{x}=\vec{b}$?

   $3\times 2$. Three equations (rows) by two variables (columns).

3. What does each row of $A$ correspond to when writing a system as $A\vec{x}=\vec{b}$?

   One equation in the system. Each row of $A$ represents the coefficients from one equation.

4. What is the variable vector $\vec{x}$ for a system in variables $x$ and $y$ written as $A\vec{x}=\vec{b}$?

   $\begin{bmatrix}x\\y\end{bmatrix}$. Variables $x$ and $y$ in column form for matrix multiplication.

5. What is the matrix equation for the system $-2x+5y=9$ and $0x-3y=-6$?

   $\begin{bmatrix}-2&5\\0&-3\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}9\\-6\end{bmatrix}$. First equation gives row $(-2,5)$, second gives row $(0,-3)$.

6. What is the variable vector $\vec{x}$ for a system in variables $x$, $y$, and $z$ written as $A\vec{x}=\vec{b}$?

   $\begin{bmatrix}x\\y\\z\end{bmatrix}$. All three variables $x$, $y$, $z$ stacked vertically.

7. What matrix equation represents the system $x-y=0$, $2x+3y=7$, $-x+4y=1$ as $A\vec{x}=\vec{b}$?

   $\begin{bmatrix}1&-1\\2&3\\-1&4\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}0\\7\\1\end{bmatrix}$. Three equations in two variables creates $3\times 2$ coefficient matrix.

8. What does the coefficient matrix $A$ contain when writing $A\vec{x}=\vec{b}$ for a linear system?

   All coefficients of the variables, arranged by equations. Each row contains coefficients from one equation in the system.

9. What does the constant vector $\vec{b}$ represent in $A\vec{x}=\vec{b}$?

   The right-hand constants stacked in a column. Column vector containing constants from the right side of each equation.

10. Find the coefficient row for $y-3x=2$ when variable order is $x,y$.

    $\begin{bmatrix}-3&1\end{bmatrix}$. Coefficients $(-3,1)$ when equation is written as $-3x+y=2$.

11. What is the coefficient matrix $A$ for $-2x+5y=9$ and $-3y=-6$?

    $\begin{bmatrix}-2&5\\0&-3\end{bmatrix}$. Missing $x$ term in second equation gives coefficient $0$.

12. What is the constant vector $\vec{b}$ for $-x-2y=-3$ and $3x=12$?

    $\begin{bmatrix}-3\\12\end{bmatrix}$. Right-hand constants $-3$ and $12$ from both equations.

13. Identify the matrix equation for $x+2y=7$ and $-3x+4y=1$ in the form $A\vec{x}=\vec{b}$.

    $\begin{bmatrix}1&2\\-3&4\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}7\\1\end{bmatrix}$. Coefficients: row 1 is $(1,2)$, row 2 is $(-3,4)$; constants: $(7,1)$.

14. Identify the matrix equation for $\frac{1}{2}x+2y=3$ and $x-\frac{3}{4}y=1$.

    $\begin{bmatrix}\frac{1}{2}&2\\1&-\frac{3}{4}\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}3\\1\end{bmatrix}$. Fractional coefficients are preserved in the matrix entries.

15. What is the matrix equation for the system $4x=8$ and $-2y=10$ using variable order $x,y$?

    $\begin{bmatrix}4&0\\0&-2\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}8\\10\end{bmatrix}$. Diagonal matrix with coefficients $4$ and $-2$ on the diagonal.

16. What is the constant vector $\vec{b}$ for the system $2x-3y=5$ and $4x+y=-1$?

    $\begin{bmatrix}5\\-1\end{bmatrix}$. Right-hand constants from both equations: $5$ and $-1$.

17. What does the variable vector $\vec{x}$ contain in a matrix equation $A\vec{x}=\vec{b}$?

    The variables stacked in a column, such as $\begin{bmatrix}x\\y\end{bmatrix}$. Column vector with all variables from the system in order.

18. What is the matrix equation form of a linear system with coefficient matrix $A$, variable vector $\vec{x}$, and constant vector $\vec{b}$?

    $A\vec{x}=\vec{b}$. Standard form where $A$ multiplies variable vector $\vec{x}$ to equal constant vector $\vec{b}$.

19. What is the coefficient matrix $A$ for $-x-2y=-3$ and $3x=12$?

    $\begin{bmatrix}-1&-2\\3&0\end{bmatrix}$. First row $(-1,-2)$, second row $(3,0)$ from the equations.

20. What is the augmented matrix corresponding to $A\vec{x}=\vec{b}$ if $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$ and $\vec{b}=\begin{bmatrix}e\\f\end{bmatrix}$?

    $\left[\begin{array}{cc|c}a&b&e\\c&d&f\end{array}\right]$. Augmented matrix combines $A$ and $\vec{b}$ with vertical separator.

21. What must you do first to write a system as $A\vec{x}=\vec{b}$ when equations are not aligned by variables?

    Rewrite each equation in the same variable order. Ensures coefficient matrix has consistent variable ordering across rows.

22. Identify the matrix equation for $2x+z=3$ and $-x+4y=5$ using variable order $x,y,z$.

    $\begin{bmatrix}2&0&1\\-1&4&0\end{bmatrix} \begin{bmatrix}x\\y\\z\end{bmatrix} = \begin{bmatrix}3\\5\end{bmatrix}$. Missing $y$ in first equation and missing $z$ in second get coefficient $0$.

23. What is the coefficient row in $A$ for the equation $-x+4y=5$ using variable order $x,y,z$?

    $\begin{bmatrix}-1&4&0\end{bmatrix}$. Coefficients $(-1,4,0)$ for variables $x$, $y$, $z$ respectively.

24. What is the defining feature of a homogeneous linear system in matrix form $A\vec{x}=\vec{b}$?

    $\vec{b}=\vec{0}$. Zero vector on right side makes the system homogeneous.

25. What is the coefficient row in $A$ for the equation $2x+z=3$ using variable order $x,y,z$?

    $\begin{bmatrix}2&0&1\end{bmatrix}$. Coefficients $(2,0,1)$ for variables $x$, $y$, $z$ respectively.

26. What is the constant vector $\vec{b}$ for $-2x+5y=9$ and $-3y=-6$?

    $\begin{bmatrix}9\\-6\end{bmatrix}$. Constants $9$ and $-6$ from the right sides.

27. What does each column of $A$ correspond to when writing a system as $A\vec{x}=\vec{b}$?

    One variable (in the chosen variable order). Each column represents coefficients of one variable across all equations.

28. Identify $A$ and $\vec{b}$ for $\begin{bmatrix}1&2\\3&4\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}5\\6\end{bmatrix}$.

    $A=\begin{bmatrix}1&2\\3&4\end{bmatrix},\ \vec{b}=\begin{bmatrix}5\\6\end{bmatrix}$. Coefficient matrix is the $2\times 2$ portion; constant vector is the right side.

29. What is the matrix equation for $x+0y+0z=2$, $0x+y+0z=3$, $0x+0y+z=4$?

    $\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}2\\3\\4\end{bmatrix}$. Identity matrix multiplied by variables equals the constant values.

30. What coefficient should be used in $A$ for a missing variable term, such as no $y$ term in an equation?

    Use $0$ as the coefficient for that variable. Missing variables have coefficient $0$ in the matrix.

31. What is the constant vector $\vec{b}$ for $3x-y+2z=4$ and $x+5y-z=6$?

    $\begin{bmatrix}4\\6\end{bmatrix}$. Constants $4$ and $6$ from right sides of both equations.

32. Which matrix equation matches the system $-x-2y=-3$ and $3x+0y=12$?

    $\begin{bmatrix}-1&-2\\3&0\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}-3\\12\end{bmatrix}$. Negative coefficients and missing $y$ term in second equation.

33. Which vector is always a column vector in the matrix equation form $A\vec{x}=\vec{b}$?

    $\vec{x}$ and $\vec{b}$ are column vectors. Both vectors are columns for proper matrix multiplication.

34. What is the coefficient matrix $A$ for $3x-y+2z=4$ and $x+5y-z=6$?

    $\begin{bmatrix}3&-1&2\\1&5&-1\end{bmatrix}$. Row 1: $(3,-1,2)$, row 2: $(1,5,-1)$ from equation coefficients.

35. What is the matrix equation for $7x-2y=0$ and $-5x+9y=0$ (a homogeneous system)?

    $\begin{bmatrix}7&-2\\-5&9\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix}$. Homogeneous system has all constants equal to zero.

36. What is $\vec{b}$ for $x+y+z=0$, $2x-y+3z=5$, $-x+4y-z=1$?

    $\begin{bmatrix}0\\5\\1\end{bmatrix}$. Constants $0$, $5$, $1$ from right sides of the equations.

37. What is the matrix equation for $x+y+z=0$, $2x-y+3z=5$, and $-x+4y-z=1$?

    $\begin{bmatrix}1&1&1\\2&-1&3\\-1&4&-1\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}0\\5\\1\end{bmatrix}$. Three equations in three variables forming $3\times 3$ system.

38. What system corresponds to $\begin{bmatrix}1&0&-2\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}7\end{bmatrix}$?

    $x-2z=7$. Single equation with three variables where $y$ coefficient is $0$.

39. What is the size of $A$ for a system of $2$ equations in $3$ variables written as $A\vec{x}=\vec{b}$?

    $2\times 3$. Two equations (rows) by three variables (columns).

40. What is the size of $\vec{x}$ for a system of $m$ equations in $n$ variables written as $A\vec{x}=\vec{b}$?

    $n\times 1$. $n$ variables stacked in one column.

41. What is the size of $\vec{b}$ for a system of $m$ equations written as $A\vec{x}=\vec{b}$?

    $m\times 1$. $m$ constants stacked in one column.

42. What is the size of $A$ for a system of $m$ equations in $n$ variables written as $A\vec{x}=\vec{b}$?

    $m\times n$. $m$ rows (equations) by $n$ columns (variables).

43. What is the difference between $A\vec{x}=\vec{b}$ and the augmented matrix $[A\mid\vec{b}]$?

    $A\vec{x}=\vec{b}$ is an equation; $[A\mid\vec{b}]$ is a single matrix. Matrix equation shows multiplication; augmented matrix shows combined data.

44. What is $A$ for $x+y+z=0$, $2x-y+3z=5$, $-x+4y-z=1$?

    $\begin{bmatrix}1&1&1\\2&-1&3\\-1&4&-1\end{bmatrix}$. Coefficients from all three equations arranged by rows.

45. What system of equations corresponds to $\begin{bmatrix}2&-1\\0&3\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}4\\9\end{bmatrix}$?

    $2x-y=4$ and $3y=9$. First equation: $2x-y=4$; second equation: $3y=9$.

46. What is the correct $A$ for $2x+y=5$ and $4x+3y=6$?

    $\begin{bmatrix}2&1\\4&3\end{bmatrix}$. First row $(2,1)$, second row $(4,3)$ match equation coefficients.

47. What matrix equation represents the system $ax+by=c$ and $dx+ey=f$ using $A\vec{x}=\vec{b}$?

    $\begin{bmatrix}a&b\\d&e\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}c\\f\end{bmatrix}$. First row has coefficients of first equation, second row has coefficients of second.

48. What is the matrix equation for $2x+3y=1$ and $6x+9y=3$ written as $A\vec{x}=\vec{b}$?

    $\begin{bmatrix}2&3\\6&9\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}1\\3\end{bmatrix}$. Standard form with proportional equations (second is $3$ times first).

49. What matrix equation represents $3x-y+2z=4$ and $x+5y-z=6$ as $A\vec{x}=\vec{b}$?

    $\begin{bmatrix}3&-1&2\\1&5&-1\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}4\\6\end{bmatrix}$. Two equations with three variables each, forming $2\times 3$ coefficient matrix.

50. What is the coefficient matrix $A$ for the system $2x-3y=5$ and $4x+y=-1$?

    $\begin{bmatrix}2&-3\\4&1\end{bmatrix}$. First row: $(2,-3)$, second row: $(4,1)$ from the two equations.

51. What matrix is the coefficient matrix for $x=2$, $y=3$, $z=4$ written as $A\vec{x}=\vec{b}$?

    $A=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$. Identity matrix has $1$s on diagonal, $0$s elsewhere.

52. Identify the error: using $\begin{bmatrix}2&1\\3&4\end{bmatrix}$ for $2x+y=5$ and $4x+3y=6$.

    The second row should be $\begin{bmatrix}4&3\end{bmatrix}$, not $\begin{bmatrix}3&4\end{bmatrix}$. Second equation has coefficients $(4,3)$, not $(3,4)$.

53. Identify the coefficient matrix $A$ for $4x=8$ and $-2y=10$ using variable order $x,y$.

    $\begin{bmatrix}4&0\\0&-2\end{bmatrix}$. Diagonal matrix form for separate single-variable equations.

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