# How do you solve the simultaneous equations 7a - 3b = 17  and 2a + b = 16?

Jul 23, 2015

$\left(a , b\right) = \left(5 , 6\right)$

#### Explanation:

[1]$\textcolor{w h i t e}{\text{XXXX}}$$7 a - 3 b = 17$
[2]$\textcolor{w h i t e}{\text{XXXX}}$$2 a + b = 16$

Multiply both sides of [2] by $3$ (to get the same coefficient for $b$ as in [1])
[3]$\textcolor{w h i t e}{\text{XXXX}}$$6 a + 3 b = 48$

[4]$\textcolor{w h i t e}{\text{XXXX}}$$13 a = 65$

Divide both sides by $13$
[5]$\textcolor{w h i t e}{\text{XXXX}}$$a = 5$

Substitute $5$ for $a$ in [2]
[6]$\textcolor{w h i t e}{\text{XXXX}}$$2 \left(5\right) + b = 16$

Simplify
[7]$\textcolor{w h i t e}{\text{XXXX}}$$b = 6$

Jul 23, 2015

$a = 5$

$b = 6$

#### Explanation:

We are given $7 a - 3 b = 17$ and $2 a + b = 16$

Let's get the $b$'s equal to each other but opposite in sign by multiplying $2 a + b = 16$ by $3$:

$2 a + b = 16$

$3 \cdot 2 a + 3 \cdot b = 3 \cdot 16$

$6 a + 3 b = 48$

Now, let's add $6 a + 3 b = 48$ to $7 a - 3 b = 17$

$6 a + 3 b = 48$
$7 a - 3 b = 17$

$13 a = 65$

$a = \frac{65}{13}$

$a = 5$

Now, substitute $a$ into $7 a - 3 b = 17$

$7 a - 3 b = 17$

$7 \cdot 5 - 3 b = 17$

$35 - 3 b = 17$

$- 3 b = 17 - 35$

$- 3 b = - 18$

$b = 6$

Let's check to see if our answers are correct by plugging in the values we found for both $a$ and $b$ into $2 a + b = 16$

$2 a + b = 16$

$2 \cdot 5 + 6 = 16$

$10 + 6 = 16$

$16 = 16$