Apr 10, 2017

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

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

Explanation:

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

$\text{ } x + y = 4$
$\text{+ } 2 x - y = - 2$

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

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

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

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

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

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

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

Simplify: $- 11 a = 36$

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

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

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