Forming Basic Expressions and Equations
Algebraic expressions are used to represent numerical relationships using symbols and operations. For example, if a number is 2 more than n, this relationship can be expressed as n + 2. Here, n is a variable representing an unknown number, and the expression shows how to calculate the number based on n.
Creating expressions requires understanding the relationship described and translating it into algebraic terms. For instance, "3 less than a number m" is written as m - 3.
Equations are statements that two expressions are equal. For example, if a number x increased by 5 equals 12, the equation is x + 5 = 12. Equations can be formed from word problems or relationships and are the starting point for solving unknown values.
- Identify the variable representing the unknown quantity.
- Translate words into algebraic terms using operations (+, -, ×, ÷).
- Write expressions for phrases like "more than", "less than", "times", "of".
- Form equations by setting expressions equal when conditions are given.
Solving Linear Equations with One Variable
A linear equation in one variable has the form ax + b = c, where a, b, and c are constants and x is the variable. The goal is to find the value of x that makes the equation true.
To solve, isolate x by performing inverse operations step-by-step, maintaining equality. For example, consider 3x + 4 = 10.
Step 1: Subtract 4 from both sides: 3x + 4 - 4 = 10 - 4, simplifying to 3x = 6. Step 2: Divide both sides by 3: 3x/3 = 6/3, giving x = 2. Always check by substituting back into the original equation: 3(2) + 4 = 6 + 4 = 10, which is correct.
- Use inverse operations to isolate the variable.
- Perform the same operation on both sides of the equation.
- Check the solution by substitution.
- Handle negative coefficients and constants carefully.
Solving Simultaneous Linear Equations with Two Variables
Simultaneous equations involve two equations with two variables, for example: 5 - 2x = 3(x + 7). The goal is to find values of x and y (or other variables) that satisfy both equations simultaneously.
First, simplify and rewrite each equation in a standard form, such as ax + by = c. For the example, expand the right side: 5 - 2x = 3x + 21. Rearranged: 5 - 2x - 3x = 21 → 5 - 5x = 21 → -5x = 16 → x = -16/5.
If there is a second equation, substitute x into it to find y. Alternatively, use elimination or substitution methods to solve the system. Always verify solutions by substituting both values into the original equations.
- Simplify each equation first.
- Use substitution or elimination methods.
- Isolate one variable and substitute into the other equation.
- Check solutions in both original equations.
Rearranging Formulas to Make a Variable the Subject
Rearranging a formula means rewriting it so that a chosen variable is alone on one side of the equation. This is essential for solving problems where a specific variable's value is required.
For example, given the formula A = 2l + 3w, to make l the subject, isolate l: Subtract 3w from both sides: A - 3w = 2l. Then divide both sides by 2: (A - 3w)/2 = l.
When the variable appears only once and without powers or roots, rearrangement involves inverse operations such as addition/subtraction and multiplication/division. Always perform the same operation on both sides and simplify carefully to avoid errors.
- Identify the variable to isolate.
- Use inverse operations stepwise.
- Maintain equality by performing operations on both sides.
- Simplify the expression fully.
- Check by substituting values if possible.
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Forming expressions | Translate word problems into algebraic expressions | "2 more than n" → n + 2 | Write clear expressions to model problems accurately |
| Solving single linear equations | Isolate variable using inverse operations | 3x + 4 = 10 → x = 2 | Show each step clearly and check solution |
| Solving simultaneous equations | Use substitution or elimination methods | 5 - 2x = 3(x + 7) → x = -16/5 | Simplify first, then solve systematically |
| Rearranging formulas | Make chosen variable the subject using inverse operations | A = 2l + 3w → l = (A - 3w)/2 | Present final formula clearly and verify correctness |
Active recall
Quick check
1. Express "5 less than a number m" as an algebraic expression.
m - 5, because "less than" indicates subtraction from the variable.
2. Solve the equation 4x - 7 = 9.
Add 7 to both sides: 4x = 16; divide both sides by 4: x = 4.
3. Given the simultaneous equations: x + y = 7 and x - y = 3, find x and y.
Add equations: (x + y) + (x - y) = 7 + 3 → 2x = 10 → x = 5. Substitute x = 5 into x + y = 7 → 5 + y = 7 → y = 2.
4. Rearrange the formula P = 2l + 2w to make w the subject.
Subtract 2l: P - 2l = 2w; divide both sides by 2: w = (P - 2l)/2.