Linear Systems: Definitions, Methods, and Word Problems

Slope-Intercept Form

$$y = mx + b$$

Slope

$m = \frac{\text{rise}}{\text{run}}$ or $\frac{\text{change in } y}{\text{change in } x}$

y-intercept

$b$ = $y$-intercept (where the line crosses the vertical $y$-axis, written as $(0, b)$)

Standard Form

$$Ax + By = C$$

System of Linear Equations

A collection of two or more linear equations sharing the same variables.

Solution to a System

The point $(x, y)$—or $(x, y, z)$ for 3 variables—that makes all equations in the system true at the same time.

Graphing Method

Best when both equations are already in slope-intercept form ($y = mx + b$) or easily plotted.

Substitution Method

Best when one variable is already isolated (e.g., $y = 2x - 5$ or $x = y + 3$).

Elimination Method

Best when equations are written in standard form ($Ax + By = C$).

One Solution

Lines intersect at exactly one point.

No Solution

Lines are parallel (same slope, different $y$-intercepts).

Infinitely Many Solutions

Lines are coincident (identical lines on top of each other).

3-Step Word Problem Process

Define Variables, Set Up Two Equations, Solve & Label.

Quantity & Value / Cost Equation 1

$$x + y = \text{Total Items}$$

Quantity & Value / Cost Equation 2

$$(\text{Rate}_1 \cdot x) + (\text{Rate}_2 \cdot y) = \text{Total Cost}$$

Hourly Rate / Work Equation

$$(\text{Hours Job A} \cdot x) + (\text{Hours Job B} \cdot y) = \text{Total Earnings}$$

Coin Collections Value Equation

$$0.05N + 0.10D = \text{Total Value}$$

Coin Collections Relationship Equation

$$D = 2N - 5$$

Common Mistake in Graphing

Moving the slope in the wrong direction.

Common Error in Elimination

Forgetting to multiply the constant on the right side of the equals sign.

Distributive Property

Multiply the term outside by every term inside, paying close attention to negative signs.

Verifying Answer

Plug both values back into both original equations to check if both yield true statements.