Solutions and Elementary Operations
Chapter 1 • Section 1-1
Solutions of Linear Equations
Example
Find all solutions of the linear equation in one variable: $$ax=b$$
Solution:
- If $a \neq 0$, there is a unique solution $x=b/a$.
- Else if $a = 0$ and $b \neq 0$, there is no solution.
- Else if $a = 0$ and $b = 0$, there are infinitely many solutions (any $x \in \mathbb{R}$).
🧠 Quick Quiz Win $10
How many solutions does the equation $0x = 5$ have?
Definitions
A linear equation is an expression of the form: $$ a_1 x_1 + a_2 x_2 + \dots + a_n x_n = b $$ where $a_i$ are real numbers (not all zero) and $b$ is a real number.
A system of linear equations is a set of $m \geq 1$ linear equations.
A solution is an $n$-tuple that satisfies each equation.
To solve a system means to find all solutions.
Example System
Consider the system:
$$ \begin{aligned} x_1 - 2x_2 - 7x_3 &= -1 \\ -x_1 + 3x_2 + 6x_3 &= 0 \end{aligned} $$Variables: $x_1, x_2, x_3$
Coefficients: $1, -2, -7$ and $-1, 3, 6$
Constants: $-1, 0$
Verifying Solutions
Is $(-3, -1, 0)$ a solution?
-(-3) + 3(-1) + 6(0) = 3 - 3 = 0 ✅
Yes! It satisfies both equations.
Is $(-1, 0, 0)$ a solution?
-(-1) + 0 + 0 = 1 ≠ 0 ❌
No! It fails the second equation.
Consistent System: A system that has at least one solution.
Inconsistent System: A system with no solutions (e.g., $x+y=0, x+y=1$).
Parametric Solutions
When a system has infinitely many solutions, we can't list them all! Instead, we describe them using parameters (like $s, t$).
Example
Consider the equation: $x - y = 2$.
We can choose any value for $y$. Let $y = t$ (where $t$ is any real number).
Then $x = 2 + t$.
Here, $t$ is the parameter. For each value of $t$, we get a specific solution (e.g., $t=0 \implies (2,0)$, $t=1 \implies (3,1)$).
🤔 Parameter Check Win $15
If $x + 2y = 0$, and we let $y = s$, what is $x$?
🧠 Quick Quiz Win $10
If a system has a solution $(1, 2)$ but fails for $(3, 4)$, is it consistent?
Geometric Interpretation
For a system of 2 variables, each equation represents a line. The solution is the intersection.
Rearranging:
$y = -x + 3$
$y = x + 5$
Intersection: $(-1, 4)$
Intersecting
One unique solution
Consistent
Parallel
No solutions
Inconsistent
Coincident
Infinitely many solutions
Consistent
Elementary Operations
To solve systems, we transform them into simpler, equivalent systems (same solutions) using three operations:
Interchange two equations
$r_1 \leftrightarrow r_2$
Multiply an equation by a nonzero number
$k \cdot r_i$ (where $k \neq 0$)
Add a multiple of one equation to another
$r_i + k \cdot r_j \to r_i$
Theorem
Performing elementary operations results in an equivalent system. The solutions do not change!
The Augmented Matrix
We can represent a system compactly using a matrix.
System
$$ \begin{aligned} x_1 - 2x_2 - 7x_3 &= -1 \\ -x_1 + 3x_2 + 6x_3 &= 0 \end{aligned} $$Augmented Matrix
$$ \left[\begin{array}{rrr|r} 1 & -2 & -7 & -1 \\ -1 & 3 & 6 & 0 \end{array}\right] $$We perform Elementary Row Operations on the matrix, which correspond exactly to the operations on equations.
Back Substitution
Once a system is in a simple "triangular" form, we can solve it easily using Back Substitution.
Consider this system:
$$ \begin{aligned} x + 2y + z &= 3 \\ y - z &= 2 \\ z &= 1 \end{aligned} $$We already know $z = 1$.
Substitute $z=1$ into the second equation:
$y - 1 = 2 \implies \mathbf{y = 3}$.
Substitute $y=3, z=1$ into the first equation:
Solution: $(-4, 3, 1)$
🧠 Final Challenge Win $20
Which of the following is NOT a valid elementary row operation?