# How do you solve the simultaneous equations 3x + 5y = 11 and 2x + y = 3?

Jul 22, 2015

I found:
$x = \frac{4}{7}$
$y = \frac{13}{7}$

#### Explanation:

From the second equation you can isolate $y$ as:
$y = 3 - 2 x$
substitute nto the first to find $x$:
$3 x + 5 \left(\textcolor{red}{3 - 2 x}\right) = 11$
$3 x + 15 - 10 x = 11$
$- 7 x = - 4$
$x = \frac{4}{7}$
substitute back into the second equation:
$y = 3 - 2 \left(\frac{4}{7}\right) = \frac{21 - 8}{7} = \frac{13}{7}$

Jul 22, 2015

$\left(x , y\right) = \left(\frac{4}{7} , \frac{13}{7}\right)$

#### Explanation:

[1]$\textcolor{w h i t e}{\text{XXXX}}$$3 x + 5 y = 11$
[2]$\textcolor{w h i t e}{\text{XXXX}}$$2 x + y = 3$

subtract $2 x$ from both sides of [2] to isolate $y$ on the left side
[3]$\textcolor{w h i t e}{\text{XXXX}}$$y = 3 - 2 x$

substitute $\left(3 - 2 x\right)$ for $y$ in [1]
[4]$\textcolor{w h i t e}{\text{XXXX}}$$3 x + 5 \left(3 - 2 x\right) = 11$

simplify
[5]$\textcolor{w h i t e}{\text{XXXX}}$$- 7 x + 15 = 11$

[6]$\textcolor{w h i t e}{\text{XXXX}}$$x = \frac{4}{7}$

substituting $\frac{4}{7}$ for $x$ in [2]
[7]$\textcolor{w h i t e}{\text{XXXX}}$$2 \left(\frac{4}{7}\right) + y = 3$

[8]$\textcolor{w h i t e}{\text{XXXX}}$$y = \frac{13}{7}$