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## Basic Skills in Higher Mathematics

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**Basic Skills in Higher Mathematics**Mathematics 1(H)Outcome 1 Robert GlenAdviser in Mathematics**Mathematics 1(Higher)**Outcome 1 Use the properties of the straight line Straight lines**Mathematics 1(Higher)**Outcome 1 Use the properties of the straight line PC Index Click on the one you want PC(a) Gradients and straight lines PC(b) Gradients and angles PC(c) Parallel and perpendicular**Mathematics 1(Higher)**Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient Gradients and straight lines**Mathematics 1(Higher)**Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient Index Click on the section you want 1 What is gradient? 2 The gradient of a line 3 The equation of a line given its gradient and the intercept on the y - axis 4 The equation of a line given one point on the line and the gradient 5 The equation of a line given two points on the line**Mathematics 1(Higher)**Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient Section 1 1 What is gradient?**Mathematics 1(Higher)**1.1 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 1 What is gradient? The gradient (slope) of this roof is 2m 2 2m = 3m 3 3m**Mathematics 1(Higher)**1.2 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 1 What is gradient? 3m The gradient (slope) of this roof is 3m 3m = 1 3m**Mathematics 1(Higher)**1.3 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 1 What is gradient? The gradient (slope) of this roof is 3m 3m 3 7m = 7m 7**Mathematics 1(Higher)**1.4 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 1 What is gradient? 3m 3 Gradient = 7 7m Check this:The steeperthe slope, the greater the gradient. 2m 2 Gradient = 3 3m 3m Gradient = 1 3m**Mathematics 1(Higher)**1.5 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 1 What is gradient? What is the gradient of this roof ? 5m 5 4 3m A B 4 5 4m 3 3 C D 4 5**Mathematics 1(Higher)**1.6 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 1 What is gradient? What is the gradient of this roof ? 5m 5 4 3m A B 4 5 4m 3 3 C D 4 5 Click on the letter of the correct answer**Mathematics 1(Higher)**1.7 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 1 What is gradient? Sorry, wrong answer What is the gradient of this roof ? 5m 5 4 3m A B 4 5 4m 3 3 Gradient = vertical horizontal C D 4 5 Have another go!**Mathematics 1(Higher)**1.8 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 1 What is gradient? What is the gradient of this roof ? 5m 5 4 3m A B 4 5 4m 3 3 C D 4 5 Click on the letter of the correct answer**Mathematics 1(Higher)**1.9 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 1 What is gradient? What is the gradient of this roof ? 5m 5 4 3m A B 4 5 4m 3 3 C D Correct! 4 5 End of Section 1**Mathematics 1(Higher)**Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient Section 2 2 The gradient of a line**Mathematics 1(Higher)**2.1 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient y 2 The gradient of a line uphill B Read all lines from left to right A Line AB is uphill from left to right x Line AB has a positive gradient mAB 0**Mathematics 1(Higher)**2.2 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient y 2 The gradient of a line P uphill B Read all lines from left to right A downhill Line PQ is downhill from left to right Q x Line PQ has a negative gradient mPQ 0**Mathematics 1(Higher)**2.3 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient y 2 The gradient of a line P uphill B Read all lines from left to right A downhill Q x Line PQ has a negative gradient mPQ 0 Line AB has a positive gradient mAB 0**Mathematics 1(Higher)**2.4 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient y 2 The gradient of a line uphill B (9, 6) change in y Gradient = change in x 3 A 39 (0, 3) mAB = 9 13 = x**Mathematics 1(Higher)**2.6 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient y 2 The gradient of a line uphill B (9, 6) change in y Gradient = 1 change in x 3 1 A 3 1 39 (0, 3) mAB = 3 13 = x Note: we could have measured the gradient like this**Mathematics 1(Higher)**2.7 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient y 2 The gradient of a line (0, 7) P change in y Gradient = change in x downhill -6 -6 9 mPQ = Q (9, 1) 23 = - 9 x**Mathematics 1(Higher)**2.8 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient y 2 The gradient of a line (0, 7) P change in y -2 Gradient = change in x 3 downhill -2 -6 9 mPQ = 3 -2 Q (9, 1) 23 3 = - x Note: we could have measured the gradient like this**Mathematics 1(Higher)**2.9 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient y 2 The gradient of a line uphill B (9, 6) change in y Gradient = change in x 6 - 3 A 6 - 3 9 - 0 (0, 3) mAB = 9 - 0 39 = x 13 =**Mathematics 1(Higher)**2.10 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient y 2 The gradient of a line (0, 7) P change in y Gradient = change in x downhill 1 - 7 1 - 7 9 - 0 mPQ = Q (9, 1) -6 9 = 9 - 0 x 23 - =**Mathematics 1(Higher)**2.11 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 2 The gradient of a line y B (x2 , y2) A formula to memorise A (x1 , y1) y2 - y1x2 - x1 x mAB =**Mathematics 1(Higher)**2.12 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 2 The gradient of a line y B (x2 , y2) A formula to memorise A (x1 , y1) y2 - y1x2 - x1 x mAB =**Mathematics 1(Higher)**2.13 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 2 The gradient of a line y B (6 ,5) 1Calculate the gradient of line AB y2 - y1x2- x1 A (2 , 3) mAB = x 5 - 36 - 2 = 24 = Did you getthis answer? 12 =**Mathematics 1(Higher)**2.14 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 2 The gradient of a line y 2Calculate the gradient of line CD. D (6 ,2) y2 - y1x2- x1 mCD = x C (2 , -1) 2 - (-1)6 - 2 = 34 Did you getthis answer? =**Mathematics 1(Higher)**2.15 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 2 The gradient of a line y E (-3 , 3) 3Calculate the gradient of line EF. y2-y1x2- x1 mEF = x F (5,-1) -1 - 35 - (-3) = -48 Did you getthis answer? = 12 = - End of Section 2**Mathematics 1(Higher)**Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient Section 3 3 The equation of a linegiven its gradient and theintercept on the y - axis**Mathematics 1(Higher)**3.1 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 3 The equation of a line given gradient and intercept y Find the equation of line KL which has a gradient of ½ and passes through the point (0, 3). (x, y) L (0,3) m = ½ K y - 3x - 0 12 mKL= = O x y - 3 = ½ x The equation of KL isy = ½ x + 3 y = ½ x + 3**Mathematics 1(Higher)**3.2 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 3 The equation of a line given gradient and intercept y Find the equation of line KL which has a gradient of ½ and passes through the point (0, 3). (x, y) L (0,3) m = ½ K Formula:y = m x + c O x The equation of KL isy = ½ x + 3**Mathematics 1(Higher)**3.3 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 3 The equation of a line given gradient and intercept y (x, y) L The equation of line with gradient m and intercept c is: (0, c) m K y = m x + c O x Memorise this**Mathematics 1(Higher)**3.4 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 3 The equation of a line given gradient and intercept y P 1 Find the equation of line PQ which has a gradient of -2 and passes through the point (0, 5). (0,5) m = -2 (x, y) Use the formula O x Q The equation of PQ isy = -2 x + 5**Mathematics 1(Higher)**3.5 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 3 The equation of a line given gradient and intercept F y (x, y) 2 Find the equation of line EF which has a gradient of ¾ and passes through the point (0, -3). O x m = ¾ (0,-3) Use the formula E The equation of EF is y = ¾ x - 3**Mathematics 1(Higher)**3.6 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 3 The equation of a line given gradient and intercept End of Section 3 You should now do Section A1 questions 1 - 10 on page 3 of the Basic Skills booklet.**Mathematics 1(Higher)**Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient Section 4 4 The equation of a linegiven one point on the line and the gradient**Mathematics 1(Higher)**4.1 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 4 The equation of a line given one point and the gradient y L Find the equation of the linethrough the point (4, 3) with gradient 3. (x, y) m = 3 (4,3) y - 3x - 4 mKL= = 3 O x y - 3 = 3(x - 4) K y - 3 = 3x - 12 The equation of KL is y = 3x - 9 y = 3x - 9**Mathematics 1(Higher)**4.2 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 4 The equation of a line given one point and the gradient y L Find the equation of the linethrough the point (4, 3) with gradient 3. (x, y) m = 3 (4,3) O x Formula:y - b = m (x - a) K The equation of KL is y = 3x - 9**Mathematics 1(Higher)**4.3 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 4 The equation of a line given one point and the gradient y L (x, y) The equation of the linethrough the point (a,b) with gradient m is : m (a, b) y - b = m (x - a) O x K Memorise this**Mathematics 1(Higher)**4.4 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 4 The equation of a line given one point and the gradient y Q 1Find the equation of the linethrough the point (-1, 2) with gradient 2. (x, y) m = 2 (-1,2) Use the formula x P O The equation of PQ is y = 2 x + 4**Mathematics 1(Higher)**4.5 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 4 The equation of a line given one point and the gradient y Q 1Find the equation of the linethrough the point (-1, 2) with gradient 2. (x, y) (a, b) m = 2 y - b = m (x - a) (-1,2) y - 2 = 2 (x - (-1)) x P y - 2 = 2 (x + 1) O y -2 = 2 x + 2 The equation of PQ is y = 2 x + 4 y = 2 x + 4**Mathematics 1(Higher)**4.6 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 4 The equation of a line given one point and the gradient y 2Find the equation of the linethrough the point (6, -2) with gradient ½. N m = ½ O x (6, -2) M (x, y) Use the formula The equation of MN is 2y = x - 10**Mathematics 1(Higher)**4.7 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 4 The equation of a line given one point and the gradient y 2Find the equation of the linethrough the point (6, -2) with gradient ½. N m = ½ O (x, y) x (6, -2) y - b = m (x - a) M y - (-2) = ½ (x - 6) (a, b) y +2 = ½(x - 6) Multiply both sides by 2to clear the fraction. 2y + 4 = x - 6 The equation of MN is 2y = x - 10 2y = x - 10 orx - 2y -10=0**Mathematics 1(Higher)**4.8 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 4 The equation of a line given one point and the gradient y R 3 Find the equation of the linethrough the point (-1, 4) with gradient 2/3 . m = -2/3 (-1,4) (x, y) S O x Use the formula The equation of RS is 3y = -2x + 10**Mathematics 1(Higher)**4.9 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 4 The equation of a line given one point and the gradient y R 3 Find the equation of the linethrough the point (-1, 4) with gradient 2/3 . m = -2/3 (-1,4) (x, y) (a, b) y - b = m (x - a) O x S y -4 = (x - (-1)) -2/3 y- 4 = -2/3 (x + 1) Multiply both sides by 3to clear the fraction. 3y - 12 = -2(x + 1) The equation of RS is 3y = -2 x + 10 3y = -2 x + 10 or 2 x + 2y -10=0**Mathematics 1(Higher)**4.9 Mathematics 1(Higher) 4.10 Outcome 1 Use the properties of the straight line Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 4 The equation of a line given one point and the gradient End of Section 4 You should now do Section A1 questions 11 - 20 on page 3 of the Basic Skills booklet.**Mathematics 1(Higher)**Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient Section 5 5 The equation of a linegiven two points on the line**Mathematics 1(Higher)**5.1 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 5 The equation of a line given two points on the line y B (6, 4) Find the equation of the linejoining the points A (3, 1) and B (6, 4) . (a, b) (3,1) m = 1 A x O Step 1 Calculate the gradient Step 2 Calculate the equation y2 - y1x2- x1 4 - 16 - 3 y - b = m (x - a) mAB = = Choose A (3, 1) as thepoint on the line.i.e. a = 3, b = 1 y - 1 = (x - 3) 1 33 = y - 1 = x - 3 (You get exactly thesame answer if youchoose B.) = 1 y = x - 2