Number and algebra basics

Key terms

Standard steps

Signs (directed numbers)

1. Change "minus a negative" to "plus": a − (−b) = a + b.
2. Multiply or divide: same signs give +, different signs give −.
3. Do brackets and powers first; (−3)² = 9 but −3² = −9.

Why: subtracting a debt is the same as gaining money.

Fractions

1. To add or subtract, make the denominators the same (use the L.C.M.).
2. To multiply, multiply tops and bottoms; cancel first if you can.
3. To divide, multiply by the reciprocal.

Solve a linear equation

1. Expand brackets.
2. Move x-terms to one side and numbers to the other. Change the sign when a term crosses the "=".
3. Divide by the coefficient of x.
4. Check by substituting back.

Why: an equation is a balance; whatever you do to one side, do to the other.

Solve simultaneous equations

1. Choose the method:
   • substitution, if one equation already gives x or y alone;
   • elimination, otherwise. Multiply so one unknown has the same coefficient, then add or subtract.
2. Solve for one unknown.
3. Substitute back to find the other.
4. Check in both equations.

Change the subject

1. Treat the new subject as the unknown.
2. Undo operations in reverse order: brackets and fractions first, then add/subtract, then multiply/divide.
3. Write the new subject on the left.

Algebraic fractions

1. Factorise every denominator.
2. Use the L.C.M. of the denominators as the common denominator.
3. Combine the numerators, keeping brackets around each one.
4. Expand, simplify, and cancel common factors.

Ratios a : b : c

1. Make the shared term the same. For a : b = 2 : 3 and b : c = 4 : 5, make b equal to 12.
2. a : b = 8 : 12 and b : c = 12 : 15, so a : b : c = 8 : 12 : 15.
3. Check by simplifying back: 8 : 12 = 2 : 3 and 12 : 15 = 4 : 5.

Errors in measurement

1. Maximum absolute error = ½ × the smallest unit of measurement.
2. Relative error = maximum absolute error ÷ measured value.
3. Percentage error = relative error × 100%.
4. The true value lies between (measured − max error) and (measured + max error).

Want to check an answer? Fraction calculator shows every step.

Worked examples

3x − 6 = x + 8      expand
3x − x = 8 + 6      move terms
    2x = 14
     x = 7          check: 3(5) = 15 = 7 + 8 ✓

x = y + 1                      from the second equation
2(y + 1) + 3y = 12             substitute
       5y + 2 = 12
            y = 2,  x = 3      check: 2(3) + 3(2) = 12 ✓, 3 − 2 = 1 ✓

= (x + 2)/[(x − 1)(x + 2)] − (x − 1)/[(x − 1)(x + 2)]
= [(x + 2) − (x − 1)] / [(x − 1)(x + 2)]
= 3 / [(x − 1)(x + 2)]

v − u = at
    a = (v − u)/t

(3x + 2y) + (5x − 2y) = 16 + 8      add: the y-terms cancel
                   8x = 24
                    x = 3
3(3) + 2y = 16                      substitute back
       2y = 7
        y = 3.5                     check: 5(3) − 2(3.5) = 15 − 7 = 8 ✓

2/3 + 3/4  = 8/12 + 9/12 = 17/12 = 1 5/12      L.C.M. of 3 and 4 is 12
5/6 ÷ 10/9 = 5/6 × 9/10                        multiply by the reciprocal
           = 45/60 = 3/4

−3²     = −9          only the 3 is squared
(−2)³   = −8          so − (−8) = + 8
(−4)(−5) = 20         same signs give +
−9 + 8 + 20 = 19

b is 3 parts and 4 parts; L.C.M. is 12
a : b = 2 : 3  = 8 : 12
b : c = 4 : 5  = 12 : 15
a : b : c = 8 : 12 : 15

Common mistakes

• (−3)² written as −9.
• −(x − 3) written as −x − 3 instead of −x + 3.
• Forgetting to multiply every term when clearing fractions.
• In algebraic fractions, writing −(x − 1) as −x − 1 when the brackets are dropped.
• Giving only one unknown in simultaneous equations.
• Rounding too early. Keep full calculator values until the last line.

How the DSE tests it

• A(1) often has a 3-mark algebraic-fraction or change-of-subject question.
• Simultaneous equations appear inside word problems, for example two ratios.
• Paper 2 has quick questions on significant figures, ratios and errors.
• These skills are used inside almost every longer question.

Questions you might not ask in class

More Form 3 maths topics