Exhaustive Algebra Guide: Changing the Subject of a Formula

General Guidance and Equipment for Changing the Subject

Required Equipment:     * Standard Writing Pen.

Procedural Guidance:     1. Individual questions must be read with extreme care prior to beginning an answer.     2. Self-verification is required to ensure that final answers are mathematically logical.     3. Full mathematical workings must be displayed for every calculation.

Original Source Reference:     * Document: Corbettmaths 2022 "Changing the Subject" Video 7.     * Website: www.corbettmaths.com/contents.

Fundamental Rearranging of Linear Equations

Example 1: Addition-Based Rearrangement     * Formula: $e = d + 5$     * Target Subject: $d$     * Transformation: Subtract $5$ from both sides.     * Result: $d = e - 5$

Example 2: Simple Multiplicative Rearrangement     * Formula: $y = 3x$     * Target Subject: $x$     * Transformation: Divide both sides by $3$.     * Result: $x = rac{y}{3}$

Example 3: Subtraction and Addition Alternatives     * Formula: $a = c - w$     * Target Subject: $c$     * Options provided in exercise:         * $c = a - w$         * $c = w - a$         * $c = aw$         * $c = a + w$ (Correct Transformation)

Example 4: Basic Addition with Constant Sums     * Formula: $x + y = 1$     * Target Subject: $y$     * Result: $y = 1 - x$

Two-Step Linear Transformations

Rearranging with Multipliers and Additives (Type 1):     * Formula: $y = 3w - a$     * Target Subject: $w$     * Step 1: Add $a$ to both sides ($y + a = 3w$).     * Step 2: Divide by $3$.     * Result: $w = rac{y + a}{3}$

Rearranging with Multipliers and Additives (Type 2):     * Formula: $c = 4d + 5$     * Target Subject: $d$     * Step 1: Subtract $5$ from both sides ($c - 5 = 4d$).     * Step 2: Divide by $4$.     * Result: $d = rac{c - 5}{4}$

Working with Negative Subjects:     * Formula: $8 + c = 3 - a$     * Target Subject: $a$     * Step 1: Add $a$ to the left side and subtract $8 + c$ from the right side.     * Step 2: $a = 3 - (8 + c) = 3 - 8 - c$     * Result: $a = -5 - c$

Linear Equations Equal to Zero:     * Formula: $2x - y + 1 = 0$     * Target Subject: $x$     * Step 1: Isolate the $x$ term ($2x = y - 1$).     * Step 2: Divide by $2$.     * Result: $x = rac{y - 1}{2}$

Proportionality and Fractional Subjects

Division/Multiplication Swap:     * Formula: $s = rac{w}{a}$     * Target Subject: $w$     * Transformation: Multiply both sides by $a$.     * Result: $w = sa$

Inverting Variables in Denominators:     * Formula: $y = rac{k}{x}$     * Target Subject: $x$     * Transformation: Multiply by $x$ and divide by $y$.     * Result: $x = rac{k}{y}$

Complex Fractional Terms:     * Formula: $s = rac{hm}{4}$     * Target Subject: $m$     * Step 1: Multiply by $4$ ($4s = hm$).     * Step 2: Divide by $h$.     * Result: $m = rac{4s}{h}$

Compound Fractional Expressions:     * Formula: $t = rac{v}{4} + 1$     * Target Subject: Express $v$ in terms of $t$     * Step 1: Subtract $1$ (t - 1 = rac{v}{4}$).     * Step 2: Multiply the entire left side by 4.     * Result: v = 4(t - 1)$or$v = 4t - 4

Fractions with Grouped Numerators:     * Formula: a = rac{w - 2}{6}     * Target Subject: w     * Step 1: Multiply by 6$($6a = w - 2).     * Step 2: Add 2.     * Result: w = 6a + 2

Powers, Roots, and Pythagoras' Theorem

Basic Square and Roots:     * Formula: t = w^2     * Target Subject: w     * Result: w = ext{±} an ext{ (not applicable), correctly } w = ext{±}\sqrt{t}     * Formula: a = \sqrt{g}     * Target Subject: g     * Result: g = a^2

Cubic Equations:     * Formula: k = y^3 + a     * Target Subject: y     * Step 1: Isolate the cube (y^3 = k - a).     * Step 2: Take the cube root.     * Result: y = \sqrt[3]{k - a}

Rooted Expressions with Internal Operations:     * Formula: r = \sqrt{3w + t}     * Target Subject: t     * Step 1: Square both sides (r^2 = 3w + t).     * Step 2: Subtract 3w.     * Result: t = r^2 - 3w

Pythagoras' Theorem:     * General Formula: a^2 + b^2 = c^2     * Target Subject: a     * Step 1: a^2 = c^2 - b^2     * Step 2: Apply square root.     * Result: a = \sqrt{c^2 - b^2}

The Cube Root Inverse:     * Formula: w = \sqrt[3]{5y - 8}     * Target Subject: y     * Step 1: Cube both sides (w^3 = 5y - 8).     * Step 2: Add 8$($w^3 + 8 = 5y).     * Step 3: Divide by 5.     * Result: y = \frac{w^3 + 8}{5}

Applications in Physics and Geometry

Velocity, Acceleration, and Displacement Formulas:     * Case 1: v = u + 10t         * Sub-task A: Find v$when$u = 4$and$t = 3.             * Calculation: v = 4 + 10 \times 3 = 4 + 30 = 34         * Sub-task B: Make u the subject.             * Result: u = v - 10t         * Sub-task C: Make t the subject.             * Step 1: v - u = 10t             * Result: t = \frac{v - u}{10}     * Case 2: v^2 = u^2 + 2as         * Target Subject: s         * Step 1: Subtract u^2$($v^2 - u^2 = 2as).         * Step 2: Divide by 2a.         * Result: s = \frac{v^2 - u^2}{2a}

Geometric Perimeters:     * Context: A rectangle where P is the perimeter.     * Given Condition: Show that P = 6x + 2.     * Goal: Express x$in terms of$P.     * Step 1: P - 2 = 6x     * Step 2: Divide by 6.     * Result: x = \frac{P - 2}{6}$</h4><h4 id="6e1a03c9-9fbf-4059-a908-e35c43f248ef" data-toc-id="6e1a03c9-9fbf-4059-a908-e35c43f248ef" collapsed="false" seolevelmigrated="true">Multi-Variable Evaluation and Ratios</h4><h4 id="4d9380f5-8a46-42fd-8306-f93173171929" data-toc-id="4d9380f5-8a46-42fd-8306-f93173171929" collapsed="false" seolevelmigrated="true"><strong>Formula:$C = 4x + 5y     * Evaluation A: Find C$when$x = 9$and$y = -2.         * Calculation: C = 4(9) + 5(-2) = 36 - 10 = 26     * Rearranging for x:         * Step 1: C - 5y = 4x         * Result: x = \frac{C - 5y}{4}     * Evaluation B: Find x$when$C = 51$and$y = 3.         * Calculation: x = \frac{51 - 15}{4} = \frac{36}{4} = 9

Ratio and Proportional Forms:     * Formula: 3y = 2x     * Task A: Write y$in terms of$x         * Result: y = \frac{2x}{3}     * Task B: Write x$in terms of$y         * Result: x = \frac{3y}{2}

Circle Geometry Variables:     * Formula: ac = \pi p     * Target Subject: p     * Result: p = \frac{ac}{\pi}$</h4><h4 id="9ead53f1-a433-46d1-831a-5282f0c23bc4" data-toc-id="9ead53f1-a433-46d1-831a-5282f0c23bc4" collapsed="false" seolevelmigrated="true">Error Analysis</h4><h4 id="3fb8c963-2533-4455-8449-7c134dae0f19" data-toc-id="3fb8c963-2533-4455-8449-7c134dae0f19" collapsed="false" seolevelmigrated="true"><strong>Scenario (Question 10):</strong> Isaac is asked to rearrange$m = 3t - 8$to make$t$$ the subject.

Conceptual Exercise: Explain the mistake Isaac has made (implies analyzing incorrect steps such as subtracting instead of adding, or incorrect division of terms).