次の連立方程式を解きます。 (1) $3x - 7y = -10$ $8x - 3y = 83$ (2) $5x + 2y = 6$ $2x - 5y = -15$ (3) $x + 2y = 11$ $5y = 5 + 2x$ (4) $x = 3y + 10$ $y = 7x + 30$ (5) $-3x - 2y = -235$ $-6x + 11y = 550$ (6) $7x + 6y - 15 = 2x + 4y + 3$ $3x - 2y = 14$

はい、承知いたしました。与えられた連立方程式を解きます。

1. 問題の内容

次の連立方程式を解きます。

(1)

$3x - 7y = -10$

$8x - 3y = 83$

(2)

$5x + 2y = 6$

$2x - 5y = -15$

(3)

$x + 2y = 11$

$5y = 5 + 2x$

(4)

$x = 3y + 10$

$y = 7x + 30$

(5)

$-3x - 2y = -235$

$-6x + 11y = 550$

(6)

$7x + 6y - 15 = 2x + 4y + 3$

$3x - 2y = 14$

2. 解き方の手順

(1)

$3x - 7y = -10$ ...(1)

$8x - 3y = 83$ ...(2)

(1) * 8 - (2) * 3 より,

$24x - 56y = -80$

$24x - 9y = 249$

$-47y = -329$

$y = 7$

(1)に代入して,

$3x - 7*7 = -10$

$3x - 49 = -10$

$3x = 39$

$x = 13$

(2)

$5x + 2y = 6$ ...(1)

$2x - 5y = -15$ ...(2)

(1) * 5 + (2) * 2 より,

$25x + 10y = 30$

$4x - 10y = -30$

$29x = 0$

$x = 0$

(1)に代入して,

$5*0 + 2y = 6$

$2y = 6$

$y = 3$

(3)

$x + 2y = 11$ ...(1)

$5y = 5 + 2x$ ...(2)

(1)より, $x = 11 - 2y$

(2)に代入して,

$5y = 5 + 2(11 - 2y)$

$5y = 5 + 22 - 4y$

$9y = 27$

$y = 3$

$x = 11 - 2*3 = 11 - 6 = 5$

(4)

$x = 3y + 10$ ...(1)

$y = 7x + 30$ ...(2)

(1)を(2)に代入して,

$y = 7(3y + 10) + 30$

$y = 21y + 70 + 30$

$-20y = 100$

$y = -5$

$x = 3(-5) + 10 = -15 + 10 = -5$

(5)

$-3x - 2y = -235$ ...(1)

$-6x + 11y = 550$ ...(2)

(1) * 2 より,

$-6x - 4y = -470$ ...(3)

(2) - (3) より,

$15y = 1020$

$y = 68$

$-3x - 2(68) = -235$

$-3x - 136 = -235$

$-3x = -99$

$x = 33$

(6)

$7x + 6y - 15 = 2x + 4y + 3$ ...(1)

$3x - 2y = 14$ ...(2)

(1)を整理すると,

$5x + 2y = 18$ ...(3)

(2) + (3) より,

$8x = 32$

$x = 4$

$3(4) - 2y = 14$

$12 - 2y = 14$

$-2y = 2$

$y = -1$

3. 最終的な答え

(1) $x = 13$, $y = 7$

(2) $x = 0$, $y = 3$

(3) $x = 5$, $y = 3$

(4) $x = -5$, $y = -5$

(5) $x = 33$, $y = 68$

(6) $x = 4$, $y = -1$