Test Your Skills With a Linear Equation in Three Variables Quiz

A linear equation in three variables must have variables raised only to the power of 1, with no products or divisions of variables. Consider the options:

x + y + z = 10: All variables are linear.

x² + y + z = 5: x² is not linear.

xy + z = 3: The term xy is a product, not linear.

x/y + z = 2: Division by a variable is not linear.

Therefore, only x + y + z = 10 represents a linear equation in three variables. This forms a plane in 3D space, satisfying the properties of linear equations.

Solve x + y + z = 6, x - y + z = 4, and x + y - z = 2 for z:

Subtract the second equation from the first: (x + y + z) - (x - y + z) = 6 - 4, simplifying to 2y = 2 → y = 1.

Substitute y = 1 into x + y + z = 6: x + 1 + z = 6 → x + z = 5.

Subtract the third equation from the first: (x + y + z) - (x + y - z) = 6 - 2 → 2z = 4 → z = 2.

Thus, z = 2.

A consistent linear system in three variables has equations that intersect at a single point. For example:

Equation 1: x + y + z = 6

Equation 2: 2x - y + z = 7

Equation 3: x - 2y - z = -3

Solving this system shows all three planes meet at a single point, confirming one unique solution. Inconsistent systems have no common points, while dependent systems involve overlapping planes, leading to infinitely many solutions.

For a system of three equations, the intersection of the corresponding planes determines the solution. Consider:

Plane 1: x + y + z = 6

Plane 2: x - y + z = 4

Plane 3: x + y - z = 2

These planes intersect at a single point if consistent, form a line if dependent, or never intersect if inconsistent. When solved, the system results in a unique intersection point representing the solution.

Inconsistent systems arise when equations have no common solution. For example:

Plane 1: x + y + z = 6

Plane 2: x + y + z = 8

These planes are parallel, as they have identical coefficients for variables but different constants. Parallel planes never intersect, meaning no point satisfies both equations. This inconsistency ensures the system has no solution.

The solution to three-variable equations geometrically represents the point of intersection of three planes. For example:

Plane 1: x + y + z = 6

Plane 2: 2x - y + z = 7

Plane 3: x - 2y - z = -3

Solving, the values for x, y, and z simultaneously satisfy all equations, locating the intersection point. If planes are parallel or do not meet, no solution exists.

Factoring is used for quadratic equations like x² - 5x + 6 = 0. Linear systems like x + y + z = 6 require substitution, elimination, or matrices. For example:

Substitute values into x + y + z = 6.

Eliminate y or z using another equation.

Factoring doesn’t apply here because there are no powers or products of variables to decompose.

The rank of a coefficient matrix indicates the number of independent equations. For a system:

Matrix A =

1 1 1

2 -1 1

1 -2 -1

The rank of A is determined by row-reducing it to echelon form. If the rank of A equals the number of variables (3), all rows are linearly independent, and the system has a unique solution. If the rank is less than the number of variables, it indicates dependency among the rows, leading to infinitely many solutions. If the rank is inconsistent with the augmented matrix, the system has no solution. Thus, rank analysis provides insight into the solvability of the system.

Determinants are essential in solving linear systems using matrices. For example:

Matrix A =

1 1 1

2 -1 1

1 -2 -1

The determinant of A is calculated as follows:

det(A) = 1(-1)(-1) + 1(1)(1) + 1(2)(-2) − 1(1)(-1) − 1(2)(1) − 1(-1)(-2)

det(A) = 1 + 1 − 4 − (-1) − 2 − 2

det(A) = 3

Since det(A) is not equal to zero, the system is consistent and has a unique solution. A determinant of zero would indicate dependency among rows, resulting in no unique solution.

Solve x + y + z = 9, 2x - y + z = 7, and x - 2y - z = -3 for x:

Add equations 1 and 2: (x + y + z) + (2x - y + z) = 9 + 7 → 3x + 2z = 16.

Subtract equation 3 from equation 1: (x + y + z) - (x - 2y - z) = 9 - (-3) → 3y + 2z = 12 → y = 4 - z.

Substitute y = 4 - z into 3x + 2z = 16, solve for x:

3x + 2(2) = 16 → 3x = 12 → x = 4.