# Systems of Linear Equations

Mathematics I · Matrices and Vectors · https://tryals.app/learn/mathematics-i/systems-of-linear-equations

## Systems & Geometry

A **linear equation** uses only sums and scalar multiples, like $2x + 3y = 6$. A **system** requires several to hold simultaneously, written compactly as **$A\mathbf{x} = \mathbf{b}$** where $A$ is the matrix, $\mathbf{x}$ is the variables, and $\mathbf{b}$ is the constants.

Geometrically, each equation in two variables is a **line**. A solution is a point on all lines at once, leading to three geometric cases:

| Case | Geometry | Meaning |
| --- | --- | --- |
| **Unique** | Cross at one point | Consistent, independent |
| **None** | Parallel and distinct | Inconsistent |
| **Infinite** | Same exact line | Dependent |

**Common pitfall:** Assuming every system has a unique solution. Always check for parallel lines or redundant equations.

## Gaussian Elimination

The systematic method is **Gaussian elimination**: apply **row operations** to the **augmented matrix** $[A \mid \mathbf{b}]$ to reach row-echelon form, then **back-substitute**.

Allowed row operations (which never change the solution set):
- **Swap** two rows
- **Scale** a row by a non-zero constant
- **Add** a multiple of one row to another

| Row Result | Meaning | Solution Count |
| --- | --- | --- |
| $0 = 1$ | Contradiction | **No solution** |
| $0 = 0$ | Redundancy | **Infinite solutions** (free variables) |
| $x = c$ | Pivot found | **Unique solution** (if full rank) |

## Practice questions

6 of this lesson's 10 practice questions, with answers. The full set is in the app.

### 1. A system of two linear equations in two unknowns has exactly one solution, no solution, or infinitely many - and never, say, exactly two. Why are those the only possibilities?

A. Because two lines meet once, never, or everywhere — not twice.
B. Because the number of solutions must divide the number of unknowns, and 2 does not divide 2 appropriately.
C. Because a system with two solutions would require the equations to be nonlinear.
D. Because linear systems are solved by elimination, and elimination can only produce one answer at a time.

**Answer:** A. Because two lines meet once, never, or everywhere — not twice.

**Why:** Each equation is a line, and two lines cross once, run parallel, or coincide, if they share two points they share all of them. So the trichotomy is geometry rather than a fact about the method. It generalises: a linear solution set is empty or an affine subspace, and a subspace with more than one point has infinitely many. Nonlinearity breaks it immediately.

Page: https://tryals.app/practice/mathematics-i/systems-of-linear-equations/a-system-of-two-linear-equations-in-two-unknowns-has-exactly-one

### 2. Sort each manipulation by whether it is a legitimate row operation on a system.

**Answer:**

- Valid row operation: Swap two rows, Multiply a row by 3, Add 2 times row 1 to row 3, Replace row 2 by row 2 minus row 1
- Not a valid row operation: Multiply a row by 0, Add one column to another column

**Why:** The three legal operations are swap, scale by a NONZERO number, and add a multiple of one row to another. Scaling by 0 wipes out an equation and cannot be reversed, so it can create solutions that were not there. Column operations mix up the variables themselves and are not row operations at all.

Page: https://tryals.app/practice/mathematics-i/systems-of-linear-equations/sort-each-manipulation-by-whether-it-is-a-legitimate-row-operation-on

### 3. Three equations in three unknowns can have no solution even when no two of them contradict each other. How?

A. It cannot, pairwise consistency guarantees a common solution
B. Because inconsistency can be a property of the three together
C. Because three equations always over-determine three unknowns
D. Because it requires two of the equations to be identical

**Answer:** B. Because inconsistency can be a property of the three together

**Why:** Each pair of planes meets in a line, and the three lines can run parallel down a prism without ever sharing a point. Inconsistency is a property of the system rather than of any pair, which is why elimination is needed to detect it, checking equations two at a time will never find this.

Page: https://tryals.app/practice/mathematics-i/systems-of-linear-equations/three-equations-in-three-unknowns-can-have-no-solution-even-when-no

### 4. Order the stages of solving a linear system by Gaussian elimination.

**Answer:**

1. Write the augmented matrix of the system
2. Use row operations to clear the entries below the first pivot
3. Continue column by column until the matrix is in row echelon form
4. Inspect for an inconsistent row or for rows of all zeros
5. Back-substitute from the bottom row upward to find the solution

**Why:** Set up the augmented matrix, eliminate downward to reach echelon form, diagnose the case, then back-substitute. Reading off a solution before checking for an inconsistent row is how students end up 'solving' systems that have no solution.

Page: https://tryals.app/practice/mathematics-i/systems-of-linear-equations/order-the-stages-of-solving-a-linear-system-by-gaussian-elimination

### 5. Select every system below that is INCONSISTENT, meaning it has no solution.

A. x - y = 0 and x - y = 5
B. x = 1 and y = 2
C. x + 2y = 4 and 3x + 6y = 7
D. x + y = 1 and 2x + 2y = 2
E. x + y = 1 and x + y = 2

**Answer:** A. x - y = 0 and x - y = 5; C. x + 2y = 4 and 3x + 6y = 7; E. x + y = 1 and x + y = 2

**Why:** Options 1, 3 and 4 demand that the same expression equal two different numbers, which is impossible. Option 2 is the same equation doubled on both sides, consistent, with infinitely many solutions. Option 5 is a perfectly ordinary unique solution. Matching coefficients but mismatched constants is exactly the signature of inconsistency.

Page: https://tryals.app/practice/mathematics-i/systems-of-linear-equations/select-every-system-below-that-is-inconsistent-meaning-it-has-no

### 6. A consistent linear system with more unknowns than equations always has infinitely many solutions.

**Answer:** True

**Why:** True. Each equation can pin at most one variable, so with more unknowns than equations at least one stays free, and a free variable in a consistent system generates infinitely many solutions. The 'consistent' qualifier matters: without it the system may have none at all.

Page: https://tryals.app/practice/mathematics-i/systems-of-linear-equations/a-consistent-linear-system-with-more-unknowns-than-equations-always