Substitution

Replace a by 7, then add.

A number beside a letter means multiplication.

Multiply before subtracting.

The fraction bar means division.

Replace both letters, then use the order of operations.

Substitute zero for the letter, and keep the constant term.

1. Substitute x=0: 0^2+6.
2. Since 0^2=0, the value is 0+6=6. A zero input does not make the whole expression zero.

Square t before multiplying by the coefficient.

1. Substitute: 3\times4^2.
2. 4^2=16, then 3\times16=48.

Use brackets around the negative number.

1. Substitute: 5\times(-3)+2.
2. -15+2=-13.

Keep both minus signs before simplifying.

1. Substitute: 2-(-5).
2. Subtracting a negative adds the positive: 2+5=7.

The square and the multiple of x have different signs.

1. Substitute: (-3)^2+2\times(-3).
2. The square is 9 and the second term is -6, so the value is 9-6=3.

Put each length into its own term.

1. P=2\times7.5+2\times3.
2. The two terms are 15 and 6, so P=21 cm.

The hourly cost is multiplied by 6; the extra charge is added once.

1. C=4.5\times6+8.
2. The hourly part is 27, so C=27+8=35. The cost is £35.

Calculate the whole numerator before dividing.

1. Substitute: \frac{5+(-1)}{2}.
2. The numerator is 4, so the value is 4\div2=2.

Three letters together mean multiply all three values.

1. V=3\times2.5\times4.
2. Calculate 2.5\times4=10, then 3\times10=30. The volume is 30\text{ cm}^3.

Square the negative value and evaluate the subtracted product separately.

1. y=(-2)^2-3\times(-2)+2.
2. The square is 4 and 3\times(-2)=-6.
3. Therefore y=4-(-6)+2=4+6+2=12.

Work out ab and 2a^2 separately before subtracting.

1. ab=(-3)\times4=-12.
2. 2a^2=2\times(-3)^2=2\times9=18.
3. The expression is -12-18=-30. Squaring a makes the square positive; the subtraction in front still remains.

Calculate the bracket before multiplying by five ninths.

1. F-32=14-32=-18.
2. C=\frac59\times(-18)=5\times(-2)=-10, so the temperature is -10 °C.

The square applies only to t. Calculate the two added terms separately.

1. Substitute: s=3\times4+\frac12\times2\times4^2.
2. The first term is 12. For the second, 4^2=16 and \frac12\times2\times16=16.
3. Add the terms: s=12+16=28.

Check each term of the substitution separately. A square multiplies the whole signed number by itself.

Part (a)

1. (-5)^2=(-5)\times(-5)=25.
2. Tess’s first term should be positive. Her second term, 15, is correct.

Part (b)

1. Substitute: (-5)^2-3\times(-5).
2. This gives 25-(-15)=25+15=40.

Substitute 8 into each formula; then compare the final costs, not just the charge per mile.

Part (a)

1. A=5+1.8\times8=5+14.4=19.4.
2. Service A costs £19.40.

Part (b)

1. B=2+2.2\times8=2+17.6=19.6, so B costs £19.60.
2. The difference is 19.60-19.40=0.20. Service B costs £0.20 more.