This question tests your ability to represent a system of linear equations as a single matrix equation in the form AX = B, where A is the coefficient matrix, X is the variable vector, and B is the constant vector. Matrix form AX = B is a compact way to write entire systems: the coefficient matrix A contains all the coefficients from the left sides of equations (organized by rows for equations and columns for variables), the variable vector X lists the unknowns as a column, and the constant vector B lists the right-hand side values. When you multiply matrix A times vector X, you get the left sides of all equations, which equals vector B (the right sides). For example, the system 2x + 3y = 7 and x - y = 1 becomes [[2, 3], [1, -1]] times [x, y] = [7, 1]. Matrix multiplication recovers the original equations! To convert 4x - y = 2 and -3x + 6y = 15 to matrix form: A = [[4, -1], [-3, 6]], X = [x, y], B = [2, 15], as in choice A. Choice A correctly constructs A with proper coefficients, X in order, and B matching constants. Choice C has the constants in vector B in wrong order—the constant vector must match equation order: first equation's right side in first position, second in second; check each equation carefully when building B! Matrix equation construction recipe: (1) Label equations and variables, (2) Build A with rows from equations and columns from variables, (3) Write X as column, (4) Write B with constants in order, (5) Combine. Verify by multiplying AX. You're mastering this—stay consistent!

This question tests your ability to represent a system of linear equations as a single matrix equation in the form $AX = B$, where $A$ is the coefficient matrix, $X$ is the variable vector, and $B$ is the constant vector. Matrix form $AX = B$ is a compact way to write entire systems: the coefficient matrix $A$ contains all the coefficients from the left sides of equations (organized by rows for equations and columns for variables), the variable vector $X$ lists the unknowns as a column, and the constant vector $B$ lists the right-hand side values. When you multiply matrix $A$ times vector $X$, you get the left sides of all equations, which equals vector $B$ (the right sides). For example, the system $2x + 3y = 7$ and $x - y = 1$ becomes $$\begin{bmatrix} 2 & 3 \\ 1 & -1 \end{bmatrix}$$ times $$\begin{bmatrix} x \\ y \end{bmatrix}$$ = $$\begin{bmatrix} 7 \\ 1 \end{bmatrix}$$. Matrix multiplication recovers the original equations! For the system $3x + 2y - z = 7$ and $-x + 4y + 5z = -2$, the matrices are $A = \begin{bmatrix} 3 & 2 & -1 \\ -1 & 4 & 5 \end{bmatrix}$ (rows for equations, columns for $x, y, z$), $X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$, $B = \begin{bmatrix} 7 \\ -2 \end{bmatrix}$ (constants in equation order), so $AX = B$ matches the system perfectly. Choice A correctly constructs the coefficient matrix $A$ with proper row-column organization, variable vector $X$ in correct order, and constant vector $B$ matching the equations. Choice B has the rows of matrix $A$ swapped, putting the second equation's coefficients in the first row, which mismatches the equation order. Remember: rows must correspond to the order of equations as given—don't rearrange them! Matrix equation construction recipe: (1) Label equations (first, second, third, etc.) and variables ($x, y, z$ in consistent order), (2) Build matrix $A$: make rows from equations (row i = coefficients from equation i), make columns from variables (column j = coefficient of variable j in each equation), include 0 for missing variables, (3) Write $X$ as column vector with variables in same order as $A$'s columns, (4) Write $B$ as column vector with constants in same order as equations, (5) Combine as $AX = B$. Example: $x + 2y = 5$ and $3x - y = 7$ gives $A = \begin{bmatrix} 1 & 2 \\ 3 & -1 \end{bmatrix}$, $X = \begin{bmatrix} x \\ y \end{bmatrix}$, $B = \begin{bmatrix} 5 \\ 7 \end{bmatrix}$. Systematic! To verify your matrix equation represents the system correctly, perform the matrix multiplication $AX$ mentally or on paper: the first row of $A$ times $X$ should give the left side of equation 1, the second row times $X$ should give equation 2's left side, etc. If $$\begin{bmatrix} 2 & 3 \\ 1 & -1 \end{bmatrix}$$ times $$\begin{bmatrix} x \\ y \end{bmatrix}$$ gives $$\begin{bmatrix} 2x + 3y \\ x - y \end{bmatrix}$$, and $B = \begin{bmatrix} 7 \\ 1 \end{bmatrix}$, then matrix equation represents $2x + 3y = 7$ and $x - y = 1$. Matrix multiplication recovers the system—this verification prevents errors!

This question tests your ability to represent a system of linear equations as a single matrix equation in the form AX = B, where A is the coefficient matrix, X is the variable vector, and B is the constant vector. Matrix form AX = B is a compact way to write entire systems: the coefficient matrix A contains all the coefficients from the left sides of equations (organized by rows for equations and columns for variables), the variable vector X lists the unknowns as a column, and the constant vector B lists the right-hand side values. When you multiply matrix A times vector X, you get the left sides of all equations, which equals vector B (the right sides). For example, 4x - y = 2 and -3x + 6y = 15 becomes [[4, -1], [-3, 6]] times [x, y] = [2, 15]. Matrix multiplication recovers the original equations! To convert this system to matrix form: (1) Write A with row 1: 4 (x), -1 (y); row 2: -3, 6. So A = [[4, -1], [-3, 6]]. (2) X = [x, y]. (3) B = [2, 15]. (4) Combine: AX = B. This matrix equation is equivalent to the original system! Choice A correctly constructs A with proper coefficients (including signs), X in order, and B matching the constants. Choice C has B's constants swapped ([15, 2] instead of [2, 15]). The constant vector must match equation order: first equation's 2 first, second's 15 second. Always double-check order! Matrix equation construction recipe: (1) Label equations and variables (x, y), (2) Build A: rows from equations, columns from variables, careful with signs, (3) Write X as column matching A, (4) Write B with constants in order, (5) Combine. Systematic! To verify, as the question suggests, multiply A by X: [[4, -1]] times [x, y] gives 4x - y, equals 2; second row -3x + 6y = 15. If it produces the left-hand sides correctly, it's good—this prevents errors! You're excelling—keep practicing!

This question tests your ability to represent a system of linear equations as a single matrix equation in the form AX = B, where coefficients are explicitly shown including zeros. When coefficients are written explicitly (like 0z or 0x), this helps identify the complete coefficient matrix without missing any zero entries. To convert the system 6x - y + 0z = 3, 0x + 2y - 5z = -8, and -x + 0y + 4z = 1 to matrix form: (1) The coefficients are already explicit: row 1 has [6, -1, 0], row 2 has [0, 2, -5], row 3 has [-1, 0, 4]. (2) Build coefficient matrix A = [[6, -1, 0], [0, 2, -5], [-1, 0, 4]]. (3) Write variable vector X = [x, y, z] and constant vector B = [3, -8, 1]. (4) The complete matrix equation is [[6, -1, 0], [0, 2, -5], [-1, 0, 4]] times [x, y, z] = [3, -8, 1]. Choice A correctly transcribes all coefficients including the explicit zeros into proper matrix positions with the correct constant vector. Choice C has the constants in vector B in wrong order - it shows [3, 1, -8] instead of [3, -8, 1], swapping the second and third equations' right-hand sides. Remember: constant vector order must match equation order! Pro tip for explicit zeros: When a system shows terms like 0x or 0y, these zeros must appear in the coefficient matrix - they're not optional! The explicit notation helps prevent the common error of accidentally skipping zero coefficients. To verify, multiply each row: [6, -1, 0] times [x, y, z] = 6x - y + 0z = 6x - y = 3 ✓; [0, 2, -5] times [x, y, z] = 0x + 2y - 5z = 2y - 5z = -8 ✓; [-1, 0, 4] times [x, y, z] = -x + 0y + 4z = -x + 4z = 1 ✓!

This question tests your ability to represent a system of linear equations as a single matrix equation in the form $AX = B$, where A is the coefficient matrix, X is the variable vector, and B is the constant vector. Matrix form $AX = B$ is a compact way to write entire systems: the coefficient matrix A contains all the coefficients from the left sides of equations (organized by rows for equations and columns for variables), the variable vector X lists the unknowns as a column, and the constant vector B lists the right-hand side values. When you multiply matrix A times vector X, you get the left sides of all equations, which equals vector B (the right sides). For example, the system $2x + 3y = 7$ and $x - y = 1$ becomes $\begin{bmatrix} 2 & 3 \\ 1 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 7 \\ 1 \end{bmatrix}$. To convert the system $x + 3y = 5$, $2x - z = 1$, $y + 2z = 4$ to matrix form: (1) Write A = $\begin{bmatrix} 1 & 3 & 0 \\ 2 & 0 & -1 \\ 0 & 1 & 2 \end{bmatrix}$, inserting 0 for missing variables (e.g., no z in first, no y in second). (2) X = $[x, y, z]$. (3) B = $[5, 1, 4]$. (4) Combine as in choice A. Choice A correctly includes zeros for missing variables in A, ensuring the matrix is 3x3 for three variables. Choice B omits the z-column, making A incompatible for multiplication with $[x, y, z]$—always include columns for all variables, using 0 where absent! Matrix equation construction recipe: (1) Label equations and variables, (2) Build A with rows from equations and columns from all variables (0 if missing), (3) Write X as column in order, (4) Write B with constants, (5) Combine. Verify by multiplying AX to match left sides. Keep up the great effort!

This question tests your ability to represent a system of linear equations as a single matrix equation in the form AX = B, where A is the coefficient matrix, X is the variable vector, and B is the constant vector. Matrix form AX = B is a compact way to write entire systems: the coefficient matrix A contains all the coefficients from the left sides of equations (organized by rows for equations and columns for variables), the variable vector X lists the unknowns as a column, and the constant vector B lists the right-hand side values. When you multiply matrix A times vector X, you get the left sides of all equations, which equals vector B (the right sides). For example, the system 2x + 3y = 7 and x - y = 1 becomes [[2, 3], [1, -1]] times [x, y] = [7, 1]. Matrix multiplication recovers the original equations! For the given matrix equation [[2, -1], [3, 4]] times [x, y] = [5, 7], multiply to recover: first equation 2x - y = 5, second 3x + 4y = 7, which matches choice A. Choice A correctly identifies the system by performing the matrix multiplication to extract the equations. Choice C has the constants in vector B in wrong order or swapped— the constant vector must match equation order: first row's product equals first constant, etc.; check each carefully! To go from matrix equation to system: (1) Multiply each row of A by X to get left sides, (2) Set equal to corresponding entry in B, (3) Write as equations. Verify by plugging back: if AX equals B, it matches. You're doing wonderfully— this reverse process strengthens understanding!

This question tests your ability to represent a system of linear equations with missing variables as a single matrix equation in the form AX = B, including zeros for absent terms in A. Matrix form AX = B is a compact way to write entire systems: the coefficient matrix A contains all the coefficients from the left sides of equations (organized by rows for equations and columns for variables), the variable vector X lists the unknowns as a column, and the constant vector B lists the right-hand side values. When you multiply matrix A times vector X, you get the left sides of all equations, which equals vector B (the right sides). For example, 6x - 2y = 1, -y + 4z = -8, 3x + z = 5 becomes [[6, -2, 0], [0, -1, 4], [3, 0, 1]] times [x, y, z] = [1, -8, 5] (zeros for missing variables). Matrix multiplication recovers the original equations! To convert this system to matrix form: (1) Write A with rows for equations, columns for x, y, z: row 1: 6, -2, 0 (no z); row 2: 0, -1, 4 (no x); row 3: 3, 0, 1 (no y). So A = [[6, -2, 0], [0, -1, 4], [3, 0, 1]]. (2) X = [x, y, z]. (3) B = [1, -8, 5]. (4) Combine: AX = B. This matrix equation is equivalent to the original system! Choice A correctly includes zeros for missing variables in A, with proper signs and order, and matches B. Choice D has B's constants reordered ([1, 5, -8] instead of [1, -8, 5]). Remember: B must match the exact equation order—second equation's -8 second. Don't reorder! Matrix equation construction recipe: (1) Label equations and variables (x, y, z), (2) Build A: rows from equations, columns from variables, insert 0 for missing ones as the question reminds, (3) Write X in order, (4) Write B with constants in equation order, (5) Combine. Systematic! To verify, multiply: row 2 [0, -1, 4] times [x, y, z] gives -y + 4z = -8, matching. Check all rows—this prevents errors! You're doing fantastic with these nuances!

This question tests your ability to verify a matrix equation representation by performing matrix multiplication and checking if it produces the original system. To verify [[6, -1], [-3, 5]] times [x, y] = [2, 11] represents the given system, multiply: (1) Row 1 times X: (6)(x) + (-1)(y) = 6x - y. (2) Row 2 times X: (-3)(x) + (5)(y) = -3x + 5y. This gives [6x - y, -3x + 5y] = [2, 11], which means 6x - y = 2 and -3x + 5y = 11, exactly matching the original system. Choice A correctly verifies by showing the multiplication produces the left sides of both equations and identifying the matching right-hand side vector. Choice C incorrectly claims coefficients go in columns not rows - this is backwards! In coefficient matrices, rows correspond to equations and columns to variables. Verification is crucial: always multiply AX mentally or on paper to ensure it produces your original system - this catches transposition errors, sign mistakes, or incorrect ordering. Matrix multiplication literally reconstructs the equations, making it a perfect self-check!

This question tests your ability to verify that a matrix equation correctly represents a given system of linear equations by checking the matrix multiplication. The key to verification is performing the matrix multiplication AX and confirming it produces the left sides of the original equations, which then equal the constant vector B. To verify that [[4, -1], [-3, 6]] times [x, y] = [2, 9] represents the system 4x - y = 2 and -3x + 6y = 9: (1) Multiply row 1 by variable vector: [4, -1] times [x, y] = 4x + (-1)y = 4x - y. (2) This should equal the first constant, 2, giving 4x - y = 2 ✓. (3) Multiply row 2 by variable vector: [-3, 6] times [x, y] = -3x + 6y. (4) This should equal the second constant, 9, giving -3x + 6y = 9 ✓. (5) Both equations match the original system perfectly! Choice A correctly identifies this verification process - multiplying the matrix by the variable vector recovers the left sides of equations. Choice B discusses the determinant, which is irrelevant for verifying the matrix equation represents the system - a nonzero determinant only tells us the system has a unique solution, not whether the representation is correct. Verification strategy: Always multiply AX to check your work! Each row of A times X should give exactly the left side of the corresponding equation. If even one coefficient is wrong, the multiplication will reveal the error immediately. This multiplication check is the gold standard for verifying matrix representations of linear systems - it directly shows that the matrix equation and the original system are mathematically equivalent!

This question tests your ability to identify the coefficient matrix A from a system of linear equations written with explicit zero coefficients. Matrix form AX = B requires the coefficient matrix A to contain all coefficients from the left sides of equations, including zeros for missing variables - this ensures proper matrix dimensions and correct multiplication. For the system -x + 4y + 0z = 2, 3x - y + 5z = -1, and 2x + 0y - z = 7, the coefficient matrix A is built by: (1) Row 1 uses coefficients from equation 1: [-1, 4, 0] (note the explicit 0 for z), (2) Row 2 uses coefficients from equation 2: [3, -1, 5], (3) Row 3 uses coefficients from equation 3: [2, 0, -1] (note the 0 for missing y). Therefore A = [[-1, 4, 0], [3, -1, 5], [2, 0, -1]]. Choice A correctly identifies this coefficient matrix with all proper signs and zeros included. Choice B transposes the matrix (swaps rows and columns), putting coefficients in wrong positions - for example, it has -1, 3, 2 in the first column instead of as the first row. Remember: rows correspond to equations, columns to variables! Matrix construction tip for systems with missing variables: When a variable doesn't appear in an equation, its coefficient is 0, not blank or omitted - you must include this 0 in the matrix to maintain proper dimensions (3×3 for three variables). For example, if equation 2 were just 3x + 5z = -1 (no y term), you write it as 3x + 0y + 5z = -1, giving row [3, 0, 5] in matrix A. To verify coefficient matrix A, check that each row matches its equation: row 1 [-1, 4, 0] represents -1·x + 4·y + 0·z = -x + 4y, row 2 [3, -1, 5] represents 3x - y + 5z, and row 3 [2, 0, -1] represents 2x + 0y - z = 2x - z, perfectly matching our original system!