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Scalar Multiples and Equation Equivalence

Two equations are equivalent if they have exactly the same solution set. One key way to produce an equivalent equation is to multiply both sides by a nonzero constant $k$ . For example, $x_1 + 2x_2 = 5$ and $3x_1 + 6x_2 = 15$ (multiply by 3) describe identical lines.

This is not just algebra bookkeeping — it matters because Gaussian elimination relies on multiplying equations by scalars and adding them together. Knowing these operations preserve solutions is the entire justification for the algorithm.

Multiplying by zero is forbidden: $0 = 0$ is true for all $(x_1, x_2)$ and gives no information. Multiplying by a negative number is fine — it just flips the sign of every term.

Formal View

Theorem 1.3 — Scalar Multiples Preserve Solutions

(s_1, \ldots, s_n)

is a solution to

a_1 x_1 + \cdots + a_n x_n = b

, and

k \neq 0

, then

(s_1, \ldots, s_n)

is also a solution to

(ka_1) x_1 + \cdots + (ka_n) x_n = kb

. The two equations have identical solution sets.

Example 1.3

The equations

x_1 + 2x_2 = 5

and

-2x_1 - 4x_2 = -10

are equivalent (multiply by

-2

). Both describe the same line in

\mathbb{R}^2

Why This Matters

This theorem is the theoretical backbone of Gaussian elimination.

Row scaling in elimination algorithms is justified by this theorem
Normalizing equations (e.g., making leading coefficient 1) preserves the solution set
In numerical computing, row scaling improves numerical stability

Learning Resources

Row reduction and echelon forms

MIT OpenCourseWare

Gilbert Strang on elimination and row operations.

47 min

Quiz

Question 1

Multiplying an equation by $-3$ changes its solution set.

Question 2

Which equation is equivalent to $2x_1 - 4x_2 = 8$ ?

Common Mistakes

Multiplying only the left-hand side by $k$ and forgetting the right-hand side.
Multiplying by zero — this destroys all information and gives a trivially true $0=0$ .