Algebra Study Guide

Kinematics is the branch of physics that describes the motion of points, bodies, and systems of bodies without considering the forces that cause them to move. In GCSE Mathematics, we focus on constant speed, known as uniform velocity. When an object moves at a constant speed, the relationship between distance, speed, and time can be described by a simple formula: Distance = Speed × Time.

Let's break down this fundamental relationship:

• Distance (d): This is the total length covered by an object during its motion. It is typically measured in units like metres (m), kilometres (km), miles, or feet.
• Speed (s): This is the rate at which an object covers distance. It tells us how much distance is covered per unit of time. Common units for speed include metres per second (m/s), kilometres per hour (km/h), or miles per hour (mph).
• Time (t): This is the duration for which the motion occurs. It can be measured in seconds (s), minutes (min), or hours (h).

The core formula, d = s × t, can be rearranged to find any of the three variables if the other two are known:

1. To find Distance: d = s × t
2. To find Speed: s = d / t (Speed = Distance ÷ Time)
3. To find Time: t = d / s (Time = Distance ÷ Speed)

It's crucial to ensure that all units are consistent when using these formulae. For example, if speed is in km/h, then distance should be in km and time in hours. If units are mixed (e.g., speed in m/s and distance in km), you must convert one or more values so that all units are compatible before doing calculations. Paying attention to units helps prevent common errors in kinematics problems. This framework allows us to solve a wide range of problems involving travel and motion at constant speeds.

A linear equation in one unknown is an algebraic statement where two expressions are equal, and there's only one variable (the unknown) raised to the power of 1. The goal of 'solving' such an equation is to find the specific numerical value of that unknown that makes the equation true. For example, in the equation x + 5 = 12, 'x' is the unknown. To solve it, we need to isolate 'x' on one side of the equals sign. We do this by applying inverse operations. If a number is added to 'x', we subtract that number from both sides. If a number is multiplied by 'x', we divide both sides by that number. The cardinal rule is: whatever operation you perform on one side of the equation, you must perform the exact same operation on the other side to maintain the equality. This principle ensures that the equation remains balanced and the solution found is correct. For example, to solve x + 5 = 12, subtract 5 from both sides: x + 5 - 5 = 12 - 5, which simplifies to x = 7. Similarly, for 3y = 15, divide both sides by 3: 3y / 3 = 15 / 3, resulting in y = 5. Equations can involve multiple steps, sometimes requiring simplification of expressions on either side before isolating the variable. This systematic application of inverse operations is fundamental to solving all linear equations.