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**Extra resources for Libreremo David Harvey Spaces Of Capital Towards A Critical Geography**

**Sample text**

Find the length of the ladder to the nearest centimetre.

When the first n positive integers are added together, their sum is given by ∑-qn(n + 1). (i) (ii) (iii) 13 Demonstrate that this result holds for the case n = 5. Find the value of n for which the sum is 105. What is the smallest value of n for which the sum exceeds 1000? The shortest side AB of a right-angled triangle is x cm long. The side BC is 1 cm longer than AB and the hypotenuse, AC, is 29 cm long. Form an equation for x and solve it to find the lengths of the three sides of the triangle.

Check for yourself that this too gives the answer d = 4. 28 Before looking at other methods for solving this pair of equations, here is another example. 34 SOLUTION 1 #6 ⇒ ᭺ 2 #5 ⇒ ᭺ Adding ⇒ Giving 1 Substituting x = –1 in equation ᭺ Adding 3 to each side Dividing by 5 18x + 30y = 72 10x – 30y = –100 28x = –28 = –1 –3 + 5y = 5y = y = 12 15 3 x ⇒ ⇒ ⇒ Simultaneous equations 1 ᭺ 2 ᭺ 3x + 5y = 12 2x – 6y = –20 Solve C1 1 Therefore x = –1, y = 3. Note In this example, both equations were multiplied, the first by 6 to give +30y and the second by 5 to give –30y.