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Answer 1 · 2017-04-10T16:43:23

a) $(\frac{2}{3}, \frac{10}{3})$ $(2/3, 10/3)$

b) $a = - \frac{36}{11} = - 3 \frac{3}{11}, b = \frac{5}{11}$ $a = -36/11 = -3 3/11, b = 5/11$

Explanation:

a) For the first system of equations use elimination because you have a $+ y$ $+y$ in the first equation and a $- y$ $-y$ in the second equation. Add the two equations directly:

$x + y = 4$ $" "x + y = 4$
$+ 2 x - y = - 2$ $"+ "2x - y = -2$

$3 x = 2; x = \frac{2}{3}$ $" " 3x = 2; " "x = 2/3$

Substitute $x$ $x$ back into one of the equations to find $y$ $y$ :
$\frac{2}{3} + y = \frac{4}{1} \cdot \frac{3}{3}$ $2/3 + y = 4/1*3/3$
$y = \frac{12}{3} - \frac{2}{3} = \frac{10}{3}$ $y = 12/3 - 2/3 = 10/3$

Solution a) $(\frac{2}{3}, \frac{10}{3})$ $(2/3, 10/3)$

To check to see if this is correct, put this point into the second equation:
$\frac{2}{1} \cdot \frac{2}{3} - \frac{10}{3} = - \frac{6}{3} = - 2$ $2/1*2/3 - 10/3 = -6/3 = -2$

b) Rearrange the first equation to get $b = 2 a + 7$ $b = 2a + 7$
Substitute this equation into the second equation:
$- 5 a - 3 (2 a + 7) = 15$ $-5a -3(2a + 7) = 15$

Distribute: $- 5 a - 6 a - 21 = 15$ $-5a -6a -21 = 15$

Add like-terms: $- 11 a - 21 + 21 = 15 + 21$ $-11a -21 +21 = 15 + 21$

Simplify: $- 11 a = 36$ $-11a = 36$

Divide by $- 11 : a = - \frac{36}{11} = - 3 \frac{3}{11}$ $-11: a = -36/11 = -3 3/11$

Substitute this value into $b = 2 a + 7$ $b = 2a + 7$ to find $b$ $b$ :
$b = 2 \cdot - \frac{36}{11} + \frac{7}{1} \cdot \frac{11}{11}$ $b = 2*-36/11 + 7/1 * 11/11$
$b = - \frac{72}{11} + \frac{77}{11} = \frac{5}{11}$ $b = -72/11 + 77/11 = 5/11$

Check the answer by inputting it into the second equation:
$- \frac{5}{1} \cdot - \frac{36}{11} - \frac{3}{1} \cdot \frac{5}{11} = \frac{180}{11} - \frac{15}{11} = \frac{165}{11} = 15$ $-5/1*-36/11 - 3/1*5/11 = 180/11 - 15/11 = 165/11 = 15$

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