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Algebra - Rearranging Formulas and Changing the Subject

Grade 8IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A formula is a mathematical rule showing the relationship between different variables. The subject of a formula is the variable that stands alone, usually on the left-hand side of the equals sign, such as yy in y=mx+cy = mx + c.

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To change the subject of a formula, we use inverse operations to isolate the required variable. The goal is to get the target variable by itself.

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Common inverse operations include: Addition ↔\leftrightarrow Subtraction, Multiplication ↔\leftrightarrow Division, and Squaring ↔\leftrightarrow Square Root.

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When rearranging, it is often helpful to follow the reverse order of operations (SAMDEB: Subtraction, Addition, Multiplication, Division, Exponents, Brackets) to 'undo' the expression.

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If the target variable appears inside a bracket, you may need to expand the bracket first. If the target variable appears as a denominator, multiply both sides by that denominator to bring it to the numerator.

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If the target variable appears more than once, you may need to collect all terms with that variable on one side and then factorize.

📐Formulae

v=u+atv = u + at

A=πr2A = \pi r^2

C=59(F−32)C = \frac{5}{9}(F - 32)

E=12mv2E = \frac{1}{2}mv^2

s=ut+12at2s = ut + \frac{1}{2}at^2

💡Examples

Problem 1:

Make aa the subject of the formula: v=u+atv = u + at

Solution:

a=v−uta = \frac{v - u}{t}

Explanation:

First, subtract uu from both sides to get v−u=atv - u = at. Then, divide both sides by tt to isolate aa.

Problem 2:

Change the subject of the formula A=πr2A = \pi r^2 to rr.

Solution:

r=Aπr = \sqrt{\frac{A}{\pi}}

Explanation:

First, divide both sides by π\pi to get Aπ=r2\frac{A}{\pi} = r^2. Then, take the square root of both sides to isolate rr.

Problem 3:

Make xx the subject of the formula: y=2x+35y = \frac{2x + 3}{5}

Solution:

x=5y−32x = \frac{5y - 3}{2}

Explanation:

Multiply both sides by 55 to get 5y=2x+35y = 2x + 3. Subtract 33 from both sides to get 5y−3=2x5y - 3 = 2x. Finally, divide by 22 to isolate xx.

Problem 4:

Rearrange P=2(l+w)P = 2(l + w) to make ll the subject.

Solution:

l=P2−wl = \frac{P}{2} - w

Explanation:

First, divide both sides by 22 to get P2=l+w\frac{P}{2} = l + w. Then, subtract ww from both sides to isolate ll. Alternatively, you could expand the bracket first: P=2l+2wP = 2l + 2w, then P−2w=2lP - 2w = 2l, so l=P−2w2l = \frac{P - 2w}{2}.