Equations, Rearranging, Inequalities & Trial/Improvement

How do you solve a basic linear equation using inverse operations?

Solve by applying inverse operations in reverse order to isolate x. (e.g. 4x - 7 = 13 → 4x = 20 → x = 5).

How do you solve an equation that contains brackets?

Expand brackets first, then isolate the variable. Example: 3(2x + 5) = 27 → 6x + 15 = 27 → 6x = 12 → x = 2.

How do you solve an equation with variables on both sides?

Eliminate the term with the smaller variable coefficient first. Example: 5x + 4 = 2x + 19 → subtract 2x → 3x + 4 = 19 → x = 5.

How do you solve an equation that contains algebraic fractions?

Multiply every term on both sides by the common denominator to clear fractions before solving.

How do you rearrange a basic formula to change the subject?

Rearrange using inverse operations to isolate target variable. Example: P = 2l + 2w → P - 2l = 2w → w = (P - 2l)/2.

How do you change the subject of a formula when the target variable appears twice?

1. Clear fractions/brackets. 2. Collect ALL terms containing the target variable on one side. 3. Factorise out target variable. 4. Divide.

What is a linear equation?

An equation where variables have a power of 1 (no x² or √x), producing straight-line graphs of form y = mx + c.

What are the SSS and DSA rules for simultaneous equations?

Same Signs Subtract (SSS): Subtract equations if matching coefficients have same sign. Different Signs Add (DSA): Add equations if matching signs differ.

What are the steps to solve simultaneous equations using elimination?

1. Match coefficients of one variable. 2. Add/subtract equations to eliminate it. 3. Solve for remaining variable. 4. Substitute back to find second variable.

How do you solve simultaneous equations using substitution?

1. Rearrange one equation to get y = ... or x = ... 2. Substitute that expression into the other equation and solve.

What do you do if neither variable's coefficients match in simultaneous equations?

Multiply one or both equations by numbers so the coefficients of x or y become equal before adding or subtracting.

How do you set up and solve word problems using simultaneous equations?

Convert facts to 2 equations: e.g. "3 pens & 2 rulers cost £4.70" → 3p + 2r = 4.70, then solve system.

What is the most important rule when multiplying or dividing inequalities by a negative number?

FLIP the inequality sign (< to >, ≤ to ≥) whenever you multiply or divide both sides by a NEGATIVE number!

How do you represent strict vs inclusive inequalities on a number line?

Open circle (○) for strict inequalities (< or >). Solid circle (●) for inclusive inequalities (≤ or ≥).

What line types do you use for drawing graphical inequalities?

Dashed line for strict inequalities (< or >). Solid line for inclusive inequalities (≤ or ≥).

How do you find and shade the correct region for a graphical inequality?

1. Draw line y = mx + c. 2. Test point (0,0) in inequality. 3. If true, shade region containing (0,0); if false, shade opposite side.

What is trial and improvement and what columns must you use in the table?

Used to approximate solutions to 1 d.p. Table columns required: x | Calculation | Comment (Too High / Too Low).

Why is the midpoint test required in trial and improvement, and how do you do it?

After finding solution lies between two 1 d.p. numbers (e.g. 3.1 & 3.2), test the midpoint (3.15) to determine correct 1 d.p. rounding.