# How do you solve 1/2a+1/3b=8, 3/2a-4/3b=-4?

Dec 31, 2016

Solution is $a = 8$, $b = 12$.

#### Explanation:

The two given are equations are

$\frac{1}{2} a + \frac{1}{3} b = 8$ ...................................(1) and

$\frac{3}{2} a - \frac{4}{3} b = - 4$ ...................................(2)

Multiplying (1) by $3$ we get

$\frac{3}{2} a + \frac{3}{3} b = 24$ ...................................(3)

and subtracting (2) from (3), we get

$\frac{3}{3} b - \left(- \frac{4}{3} b\right) = 24 - \left(- 4\right)$

or $\frac{3}{3} b + \frac{4}{3} b = 24 + 4$

or $\frac{7}{3} b = 28$ i.e.

$b = 28 \times \frac{3}{7} = 4 \cancel{28} \times \frac{3}{1 \cancel{7}} = 12$

Putting $b = 12$ in (1), we get

$\frac{1}{2} a + \frac{1}{3} \times 12 = 8$

or $\frac{1}{2} a + 4 = 8$

or $\frac{1}{2} a = 8 - 4 = 4$

i.e. $a = 8$

Hence, solution is $a = 8$, $b = 12$.