Is it true that All simultaneous equations have a unique solution.?

Some simultaneous equations may have no solution (parallel lines) or infinite solutions (identical lines).

Is it true that The rules for inequalities are exactly the same as for equations.?

Multiplying or dividing an inequality by a negative number requires reversing the inequality sign.

Is it true that There's a single formula for the surface area of all prisms.?

The surface area of a prism requires calculating the area of each unique face and summing them up.

Algebraic Equations & Geometry Formulas Cheat Sheet

This cheat sheet covers fundamental algebraic techniques like solving simultaneous equations (graphing, substitution, elimination) and inequalities, alongside key geometric area and volume formulas for various shapes.

Core Principles

Simultaneous equations can be solved using graphing, substitution, or elimination methods.
Substitution involves replacing a variable in one equation with an expression from another.
Elimination requires manipulating equations so that one variable cancels out when added or subtracted.
Inequalities follow similar rules to equations, with a crucial change when multiplying or dividing by negative numbers.
Geometric area formulas depend on the shape, often involving base, height, radius, or diagonals.
Volume formulas for prisms and cylinders typically involve the area of the base multiplied by the height.

Action Steps

For substitution: Isolate one variable in one equation, then substitute into the other.
For elimination: Align equations, multiply to match coefficients if needed, then add or subtract.
When solving inequalities, remember to flip the inequality sign if multiplying or dividing by a negative number.
To find the area of composite shapes, break them down into simpler, known shapes.
When calculating surface area, ensure all faces of the 3D object are accounted for.
For volume, consistently use the area of the base and the perpendicular height.

Formulas

Area of Triangle: $A = \frac{1}{2}bh$ or $A = \frac{1}{2}ab\sin C$
Area of Rhombus/Kite: $A = \frac{1}{2}d_1d_2$
Area of Parallelogram: $A = bh$
Area of Trapezium: $A = \frac{1}{2}(a+b)h$
Area of Circle: $A = \pi r^2$
Area of Sector: $A = \frac{\theta}{360} \pi r^2$
Surface Area of Cylinder: $SA = 2\pi r^2 + 2\pi rh$
Surface Area of Sphere: $SA = 4\pi r^2$
Volume of Prism: $V = \text{Area of base} \times h$
Volume of Cylinder: $V = \pi r^2 h$
Volume of Cone: $V = \frac{1}{3} \pi r^2 h$
Volume of Sphere: $V = \frac{4}{3} \pi r^3$
Distance = Speed $\times$ Time ($d=st$)

Key Terms

Simultaneous Equations: A set of two or more equations that share the same variables.
Substitution Method: Solving simultaneous equations by substituting an expression for one variable into another equation.
Elimination Method: Solving simultaneous equations by adding or subtracting equations to eliminate one variable.
Inequality: A mathematical statement comparing two expressions using symbols like <, >, ≤, or ≥.
Coefficient: A numerical or constant quantity placed before and multiplying the variable in an algebraic expression.
Slant Height: The distance from the apex of a cone to a point on the circumference of its base.

Pro Tips

Always check your answers by substituting them back into the original equations.
When eliminating, ensure the signs of the coefficients are considered (add if different, subtract if same).
For inequalities, visualize the number line to understand 'greater than' or 'less than'.
Round final answers to the specified decimal place, but keep intermediate calculations more precise.
Don't forget units in your final answers for geometry and physics problems.

Pitfalls to Avoid

Forgetting to flip the inequality sign when multiplying or dividing by a negative.
Errors in algebraic manipulation, especially with signs.
Using the wrong formula for the specific geometric shape.
Not accounting for all surfaces when calculating surface area.
Confusing area formulas with perimeter formulas.

Myth vs Reality

All simultaneous equations have a unique solution.: Some simultaneous equations may have no solution (parallel lines) or infinite solutions (identical lines).
The rules for inequalities are exactly the same as for equations.: Multiplying or dividing an inequality by a negative number requires reversing the inequality sign.
There's a single formula for the surface area of all prisms.: The surface area of a prism requires calculating the area of each unique face and summing them up.

Real World Examples

A river flows at 2 m/s, and Brendan swims at 3 m/s.: Calculating downstream and upstream travel times and distances to determine total distance swum.
The sum of two children's ages is 17, and the difference is 5.: Using simultaneous equations to determine the individual ages of the children.
Finding the area of a composite shape made of rectangles and triangles.: Breaking the shape into individual components and summing their areas.

Statistics

1 cm³ to mL conversion: 1 cm³ = 1 mL
1 m³ to L conversion: 1 m³ = 1000 L

People

Brendan: Subject of a word problem involving speed, distance, and time.
Kara: Subject of an age-related word problem solved with simultaneous equations.
Ben: Subject of an age-related word problem solved with simultaneous equations.