U4D3 Solving Systems of Linear Equations by Elimination

Vocabulary flashcards covering methods, worked examples, and special cases for solving systems of linear equations by elimination.

Solving by Elimination

An algebraic method for solving a system of linear equations by adding or subtracting equations to eliminate one variable, allowing the remaining variable to be solved directly.

Elimination by Addition

The technique of adding two equations together when the coefficients of one variable are additive opposites (such as $-5y$ and $+5y$), causing that variable to cancel out.

Elimination by Subtraction

The technique of subtracting one equation from another when the coefficients of a variable are identical (such as $3x$ and $3x$), eliminating that variable.

Clearing Fractions

Multiplying every term in an equation by the lowest common denominator (such as multiplying by $4$ or $15$) to remove fractional coefficients before applying elimination.

Solution to Example 1

The coordinate pair $(2, 3)$, obtained by adding $3x - 5y = -9$ and $4x + 5y = 23$ to find $7x = 14$, then substituting $x = 2$ back into an equation.

Solution to Example 2

The coordinate pair $(3, -2)$, obtained by multiplying $x - 2y = 7$ by $3$ to match coefficients with $3x + 4y = 1$ and subtracting to solve for $y$.

Solution to Example 3

The coordinate pair $(1, -2)$, obtained by multiplying $2x + 3y = -4$ by $5$ and $5x - 4y = 13$ by $2$ to eliminate $x$.

Solution to Example 4

The solution $x = \frac{75}{7}$ and $y = -\frac{90}{7}$, obtained after clearing denominators in $\frac{1}{2}x - \frac{3}{4}y = 15$ and $\frac{2}{3}x + \frac{2}{5}y = 2$ and solving the resulting integer system.

Infinite Solutions (Identity)

A condition in a linear system where eliminating variables produces a true statement such as $0 = 0$ or $5 = 5$, indicating that the equations represent coincident lines.

No Solution (Contradiction)

A condition in a linear system where eliminating variables produces a false statement such as $0 \neq 5$, indicating that the lines are parallel and non-intersecting.