Chapter 1, 2: Linear Equations

PRIOR KNOWLEDGE - SAC 1, mainly tips

Simultaneous Equations: Methods to Solve

1. Substitution: solve for one variable and then substitute that expression into the other equation.

2. Elimination: Choose one of the variables to eliminate by first matching the coefficient and ± two equations for elimination.

Linear Inequalities

ONLY x/÷ a negative number will make both negative.

Inequality Symbols

• "<" or ">": Use a dashed line when graphing.
  Example: x > 3 → dashed line at x = 3, shading to the right.1

• "≤" or "≥": Use a solid line when graphing.
  Example: x ≤ 2 → solid line at x = 2, shading to the left.

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Shading Direction

• For "<" or "≤": Shade to the left of the line if the inequality is less than.

• For ">" or "≥": Shade to the right of the line if the inequality is greater than.

Midpoint formula

	=	coordinates of the midpoint
	=	coordinates of the first point
	=	coordinates of the second point

Distance (between two points on a linear equation) formula

d	=	distance
	=	coordinates of the first point
	=	coordinates of the second point

Gradient formula

m	=	slope
	=	coordinates of first point in the line
	=	coordinates of second point in the line

Alt. Gradient Formula

• Formula: Gradient = tan(θ)

• Definition: tan(θ) = Rise / Run = Δy / Δx

• Example: Rise = 4, Run = 3 → tan(θ) = 4/3 = 1.333

• Positive Gradient: 0° < θ < 90° → tan(θ) > 0

• Negative Gradient: 90° < θ < 180° → tan(θ) < 0

• Key: tan(θ) reflects line slope direction (up = +, down = −).

Forms of linear equation: Gradient-int form

y=mx+c, c = y-int, m = gradient

Forms of linear equation: Point-gradient form

y - y₁ = m(x - x₁), (x₁, y₁) is a point on the line with gradient m, and (x, y) is any other point on the line.

Forms of linear equation: Intercept form

x/a + y/b = 1, a line that has x-int at (a,0) and a y-int at (0,b)

Forms of linear equation: General form

mx+ny+p=0, m,n ≠ 0

• m: The coefficient of x, related to the slope of the line.

• n: The coefficient of y, also affecting the slope and orientation.

• p: The constant term, which shifts the line up (+) or down (-).

Other special aspects of Linear Equations

If a line is horizontal, y = c (c = y-int); if a line is vertical, x = a (a = x-int).

Parallel: m₁ = m₂; perpendicular: m₁m₂ = -1, m,n ≠ 0

Family of straight lines: change the value of c; m remains (y = mx + c).

Special cases of linear models with two lines—3 cases

1. Unique Solution (intercept at one point);

2. Infinitely many (lines coincide/overlap completely);

3. No solutions (parallel).