Algebra Revision Notes

The Core Kinematics Formula

The fundamental formula relating distance, speed, and time for motion at a constant speed is: Distance = Speed × Time. This is often written as d = s × t. This formula is applicable when an object moves without changing its speed or direction over a given period. Understanding this relationship is key to solving most kinematics problems at this level. You must remember this formula and how to rearrange it to find any of the three variables.

Rearranging the Formula

The core formula d = s × t can be rearranged to find any unknown quantity. If you need to find speed, the formula becomes s = d / t (Speed = Distance ÷ Time). If you need to find time, the formula becomes t = d / s (Time = Distance ÷ Speed). A useful way to remember these is a 'distance-speed-time triangle', where 'd' is at the top, and 's' and 't' are at the bottom. Cover the variable you want to find, and the remaining visible variables show the calculation.

Units and Conversions

One of the most common pitfalls in kinematics problems is inconsistent units. Always ensure that the units for distance, speed, and time are compatible before performing any calculations. For example, if speed is given in kilometres per hour (km/h), then distance should be in kilometres (km) and time in hours (h). If you have distance in metres and speed in km/h, you will need to convert one of them. Remember common conversions: 1 km = 1000 m, 1 hour = 60 minutes, 1 minute = 60 seconds, so 1 hour = 3600 seconds.

Applying the Formula to Problem Solving

When solving kinematics problems:

1. Identify what quantities are given (distance, speed, time) and what quantity needs to be calculated.
2. Check units. Convert if necessary to ensure consistency.
3. Choose the correct rearranged formula (d = s × t, s = d / t, or t = d / s).
4. Substitute the known values into the formula.
5. Calculate the result and state the correct units. Always think logically about the answer; does it make sense in the context of the problem?

What is a Linear Equation in One Unknown?

A linear equation in one unknown is an algebraic statement where an equality sign (=) separates two expressions. It features a single variable (the 'unknown'), which is only raised to the power of 1. For example, 3x - 7 = 5 is a linear equation in one unknown ('x'). The variable does not appear in denominators, under square roots, or as an exponent. The 'solution' to such an equation is the specific value of the unknown that makes the statement true. Our aim is to find this value. Example: 2a - 4 = 10 is a linear equation. The unknown is 'a'. We need to find the value of 'a'.

The Principle of Balance: Inverse Operations

The fundamental rule for solving linear equations is the 'principle of balance'. This means that whatever operation you perform on one side of the equals sign, you must perform the exact same operation on the other side to keep the equation true. To isolate the unknown, you use inverse operations. Addition is the inverse of subtraction, and multiplication is the inverse of division. If you have x + 3 = 8, you subtract 3 from both sides. If you have 4x = 20, you divide both sides by 4. Always work to 'undo' operations in reverse order of operations (PEMDAS/BODMAS applies when simplifying, so you 'undo' addition/subtraction before multiplication/division).

Steps to Solve Linear Equations

Follow these general steps to solve linear equations: 1. If necessary, simplify both sides of the equation by combining like terms or expanding brackets. 2. Use addition or subtraction to collect all terms involving the unknown on one side of the equation (usually the left) and all constant terms on the other side (usually the right). Remember to perform the inverse operation on both sides. 3. Use multiplication or division to isolate the unknown. If the unknown is multiplied by a number, divide both sides by that number. If it is divided, multiply both sides. 4. Check your solution by substituting the value you found back into the original equation to ensure both sides are equal. For 5x - 3 = 12: Add 3 to both sides: 5x = 15. Divide by 5: x = 3. Check: 5(3) - 3 = 15 - 3 = 12 (Correct).

Handling Negative Numbers and Fractions

Be careful when dealing with negative numbers. Subtracting a negative number is the same as adding a positive one (e.g., x - (-5) is x + 5). Similarly, multiplying or dividing by a negative number flips the sign of the result. When solving equations involving fractions, you can often clear the denominators by multiplying every term in the equation by a common multiple of the denominators. For example, in x/2 + x/3 = 5, multiply every term by 6 (the LCM of 2 and 3) to get 3x + 2x = 30, which simplifies to 5x = 30, so x = 6. This removes fractions and simplifies the solving process.

What is a Linear Equation?

A linear equation in one unknown is an equation that can be written in the form ax + b = c, where 'x' is the unknown variable, and 'a', 'b', and 'c' are known numerical values. The 'a' cannot be zero. The term 'linear' means that when graphed, the equation would form a straight line – this is because the variable 'x' is raised only to the power of 1. Such equations have exactly one solution for the unknown variable. For example, 3x - 7 = 8 is a linear equation. Solving it means finding the specific value of 'x' that makes the equation true.

The Golden Rule: Balance the Equation

The most important principle when solving any equation is to maintain balance. Whatever operation you perform on one side of the equals sign, you must perform the exact same operation on the other side. This ensures that the equation remains true and equivalent to the original one. Think of an equation as a perfectly balanced scale; if you add weight to one side, you must add the same weight to the other to keep it level. This applies to addition, subtraction, multiplication, and division. For instance, if you have x + 3 = 7, subtracting 3 from both sides gives x = 4.

Inverse Operations: Undoing the Math

To isolate the unknown variable, you need to 'undo' the operations performed on it. Each mathematical operation has an inverse: addition undoes subtraction, and subtraction undoes addition. Multiplication undoes division, and division undoes multiplication. When solving, you usually work backwards through the order of operations (BEDMAS/BODMAS). This means addressing addition and subtraction terms first, then multiplication and division. For example, to solve 4x - 6 = 10: first add 6 to both sides (4x = 16), then divide by 4 (x = 4).

Equations with Variables on Both Sides

If an equation has the unknown variable on both sides (e.g., 5x + 3 = 2x + 12), your first step should be to gather all the variable terms on one side and all the constant terms on the other. It's often easier to move the smaller variable term to avoid negative coefficients. For example, in 5x + 3 = 2x + 12, subtract 2x from both sides to get 3x + 3 = 12. Then, subtract 3 from both sides: 3x = 9. Finally, divide by 3: x = 3. Check your answer by substituting back into the original equation: 5(3) + 3 = 15 + 3 = 18 and 2(3) + 12 = 6 + 12 = 18. Since 18 = 18, the solution is correct.

Standard Steps for Solving Linear Equations

To solve a linear equation, the goal is to isolate the unknown variable. First, simplify each side of the equation. This may involve expanding brackets or combining like terms. Next, collect all terms containing the variable on one side of the equation (e.g., the left side) and all constant terms on the other side (e.g., the right side). This is typically done by adding or subtracting terms from both sides. Finally, divide both sides by the coefficient of the variable to find its value.

Dealing with Fractions in Equations

When linear equations involve fractions, a highly effective strategy is to eliminate the denominators. Identify the lowest common multiple (LCM) of all denominators in the equation. Then, multiply every single term on both sides of the equation by this LCM. This operation clears the denominators, converting the equation into a simpler form without fractions, which can then be solved using standard steps. Be careful to multiply *every* term, including any whole numbers or terms without denominators that might be present.

Expanding Brackets and Combining Terms

If an equation contains brackets, the first step should be to expand them. This means multiplying the term outside the bracket by each term inside the bracket. For example, '2(x + 3)' becomes '2x + 6'. After expanding, identify and combine any like terms on each side of the equation. Like terms are terms that have the same variable raised to the same power (e.g., '3x' and '5x' are like terms, but '3x' and '3x^2' are not). Combining them simplifies the equation before moving terms across the equals sign.

Setting up Equations from Word Problems

For word problems, the key is to translate the given information into a mathematical equation. Read the problem carefully and identify the unknown quantity. Assign a variable (e.g., 'x') to represent this unknown. Break down the problem into smaller parts and represent each piece of information algebraically. Formulate the equation based on the relationships described in the problem. Once the equation is set up, solve it using the methods described above. Always check if your solution makes sense in the context of the original problem.