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Slideshow 13, Mathematics Mr Richard Sasaki, Room 307 • Look at multiplying both sides of an equation by a number • Review the process of solving simultaneous equations • Solve simultaneous equations for unknowns with different coefficients Try the short multiplication worksheet and fill in the gaps, should take just a few minutes! 11 8 11 4 5 3 9 2 -4 (3𝑥 + 2) 3 2 4 5 2 (7𝑎 − 3) (9𝑥 + 4) 3𝑦 + 5 -2 4𝑥 – 2 = 8 −2𝑦 + 5 = −11 We can solve simultaneous equations by adding or subtracting when a term and coefficient is the same in both equations. ① 17𝑥 + 5𝑦 = 2 ② 17𝑥 − 2𝑦 = 23 ①－ ② 7𝑦 = −21 𝑦 = −3 ①17𝑥 + 5𝑦 = 2 17𝑥 + 5(−3) = 2 17𝑥 – 15 = 2 17𝑥 = 17 𝑥 = 1 But how do we solve simultaneous equations when there is no term and coefficient in common? As in the multiplication worksheet, we can multiply both sides of an equation by a number. We can make it so the same term shares the same coefficient. Example Solve the simultaneous equations below. What should we do first? So let’s subtract ① ① 3𝑥 + 2𝑦 = 4 ② 2𝑥 + 𝑦 = 3 The easiest way to make a from ②×2. pair of coefficients the same ②×2 4𝑥 + 2𝑦 = 6 ②×2 4𝑥 + 2𝑦 = 6is to make ② contain 2𝑦. So ① 3𝑥 + 2𝑦 = 4 𝑥 = 2 Let’s substitute 𝑥 = 2 into ②. we will multiply ② by 2. 2𝑥 + 𝑦 = 3 2(2) + 𝑦 = 3 So 𝑥 = 2 and 𝑦 = −1. 4+ 𝑦 = 3 𝑦 = −1 ② Example Solve the simultaneous equations below. 2𝑎 – 5𝑏 = 11 ② 3𝑎 + 2𝑏 = 7 Let’s multiply ① by 3 and ② by 2. Then we will have 6𝑎 in both equations. ① x 3 6𝑎 – 15𝑏 = 33 ① ② x 26𝑎 + 4𝑏 ① x 3 - ② x 2−19𝑏 = 14 = 19 𝑏 = −1 So 𝑎 = 3 and 𝑏 = −1. ①2𝑎 – 5(−1) = 11 Try the worksheets! 2𝑎 + 5 = 11 2𝑎 = 6 𝑎 = 3 𝑥 = 6, 𝑦 = 1 𝑎 = 3, 𝑏 = 6 𝑥 = 3, 𝑦 = 8 𝑥 = 1, 𝑦 = 4 𝑥 = 2, 𝑦 = 2 𝑥 = 2, 𝑦 = −1 𝑥 = 2, 𝑦 = −1 𝑥 = 4, 𝑦 = 3 𝑥 = 5, 𝑦 = 6 𝑥 = 10, 𝑦 = 1 𝑎 = ½ , 𝑏 = 3 𝑥 = 4, 𝑦 = −1 𝑥 = −2, 𝑦 = 1 𝑥 = 2, 𝑦 = −3 𝑥 = −2, 𝑦 = −3 𝑥 = 2, 𝑦 = 9 𝑥 = −3, 𝑦 = 3 𝑥 = 10, 𝑦 = 15 𝑥 = −5, 𝑦 = 13 𝑥 = −17, 𝑦 = 20 𝑚 = 5½, 𝑛 = 1 𝑥 = 3, 𝑦 = 4 𝑥 = 4, 𝑦 = 3 𝑥 = 1, 𝑦 = 1 𝑎 = 9½, 𝑏 = 5½ Chicken eggs: 20g Duck eggs: 35g There are 15 two yen coins and 25 five yen coins