Study guide: Equations, Circles and Linear Relationships

📋 Course Outline

1. Solving Linear Equations
2. Quadratics and Word Problems
3. Circle Parts and Core Formulas
4. Arcs, Sectors and Composite Shapes
5. Coordinates and Number Patterns
6. Straight Lines and Intersections

📖 1. Solving Linear Equations

★ Must-know

📌 An equation is solved by isolating the variable while performing the same operation on both sides.

📌 Opposite operations undo one another: addition and subtraction are opposites, while multiplication and division are opposites.

• To solve a two-step equation, undo the operations in reverse order.

📌 For variables on both sides, move all variable terms to one side and all numerical terms to the other side.

📌 The distributive law states $a(b+c)=ab+ac$, so the factor outside brackets must multiply every term inside.

Further detail

• The equation $3x+5=20$ gives $3x=15$ and then $x=5$.

💡 Memory Hook

Undo operations in reverse order: brackets, terms, coefficients.

📖 2. Quadratics and Word Problems

★ Must-know

📐 Formula — A simple quadratic equation can have the form $ax^2=c$; isolate $x^2$ and then take the square root of both sides, remembering the ± sign.

• 🔄 Substitution follows these steps:

  1. Replace the variable with its given value
  2. Use brackets for negative values
  3. Simplify the expression

• 🔄 Word problems follow these steps:

  1. Choose a variable, usually $x$
  2. Translate the words into an equation
  3. Solve the equation
  4. Check that the answer makes sense

Further detail

• The equation $4x^2=36$ gives $x^2=9$ and therefore $x=\pm3$.

• If $y=3x+2$ and $x=4$, substitution gives $y=3(4)+2=14$.

💡 Memory Hook

Isolate $x^2$ → square-root both sides → remember ±.

📖 3. Circle Parts and Core Formulas

🔑 Key Concepts & Definitions

• Radius: the distance from the centre of a circle to its circumference
• Diameter: a straight line across a circle through its centre

★ Must-know

📐 Formula — The diameter and radius satisfy $d=2r$ and $r=\frac{d}{2}$.

📐 Formula — The circumference of a circle is $C=2\pi r$ or $C=\pi d$ and is measured in normal length units such as cm, m, or mm.

📐 Formula — The area of a circle is $A=\pi r^2$ and is measured in squared units such as cm² or m².

Further detail

• Other circle parts include: a chord joining two points on the circumference, an arc as a curved part of the circumference, a sector as a slice of a circle, a semicircle as half a circle

💡 Memory Hook

Circumference measures around; area measures inside.

📖 4. Arcs, Sectors and Composite Shapes

★ Must-know

📐 Formula — The arc length of a sector is $L=\frac{\theta}{360}(2\pi r)$, where $\theta$ is the sector angle.

📐 Formula — The area of a sector is $A=\frac{\theta}{360}\pi r^2$.

📐 Formula — The perimeter of a sector is $P=2r+\text{arc length}$ because it includes two straight radii and one curved arc.

• For a composite shape, break it into simpler shapes, add areas of pieces, subtract areas of holes, and count only the outside boundary for perimeter.

Further detail

• Common sector fractions are:
  • 90° is one quarter of a circle
  • 180° is one half of a circle
  • 270° is three quarters of a circle
  • 360° is the whole circle

💡 Memory Hook

Angle ÷ 360 × whole circle; then add the two radii for sector perimeter.

📖 5. Coordinates and Number Patterns

🔑 Key Concepts & Definitions

• Coordinates: a pair written as $(x,y)$, with the x-value giving horizontal movement first and the y-value giving vertical movement second

★ Must-know

• The Cartesian plane has a horizontal x-axis, a vertical y-axis, and the origin at $(0{,}0)$.

• To find a number-pattern rule, calculate the difference between terms to obtain the coefficient of n, then determine what must be added or subtracted.

Further detail

• The quadrants have sign patterns I $(+,+)$, II $(-,+)$, III $(-,-)$, and IV $(+,-)$.

• The sequence 4, 7, 10, 13, 16 has difference +3 and rule $3n+1$.

💡 Memory Hook

Across first, then up or down: $x$ before $y$.

📖 6. Straight Lines and Intersections

★ Must-know

📐 Formula — A straight-line equation has the form $y=mx+b$, where $m$ is the gradient and $b$ is the y-intercept.

📐 Formula — The gradient is $m=\frac{\text{rise}}{\text{run}}$ or $m=\frac{y_2-y_1}{x_2-x_1}$.

• 🔄 Graphing a line follows these steps: Choose values for x, Substitute them to find y, Plot the coordinates, Join the points with a straight line

📌 To test whether a point lies on a line, substitute its x-value into the equation and check whether the result equals its y-value.

• To find the intersection of two lines, set their equations equal, solve for x, substitute that x-value into either equation to find y, and write the coordinate pair.

Further detail

• A positive gradient makes a line rise from left to right, a negative gradient makes it fall, and a zero gradient produces a horizontal line such as $y=4$.

• The lines $y=x+2$ and $y=-x+6$ intersect at $(2{,}4)$.

💡 Memory Hook

Gradient controls steepness, whereas the y-intercept gives where the line crosses the y-axis.

📊 Synthesis Tables

Circle quantities

Quantity	Formula	Meaning
Circumference	$C=2\pi r$ or $C=\pi d$	Distance around
Area	$A=\pi r^2$	Space inside
Arc length	$L=\frac{\theta}{360}(2\pi r)$	Part of circumference
Sector area	$A=\frac{\theta}{360}\pi r^2$	Fraction of circle area