# How do you solve the following system: 3x + 4y = 11 , 7x+15y=32 ?

Nov 21, 2016

$y = \frac{19}{17}$ and $x = \frac{37}{17}$

#### Explanation:

Step 1) Solve the first equation for $x$L

$3 x + 4 y - 4 y = 11 - 4 y$

$3 x + 0 = 11 - 4 y$

$3 x = 11 - 4 y$

$\frac{3 x}{3} = \frac{11 - 4 y}{3}$

$1 x = \frac{11 - 4 y}{3}$

$x = \frac{11}{3} - \frac{4 y}{3}$

Step 2) Substitute $\frac{11}{3} - \frac{4 y}{3}$ for $x$ in the second equation and solve for $y$:

$7 \cdot \left(\frac{11}{3} - \frac{4 y}{3}\right) + 15 y = 32$

$\frac{77}{3} - \frac{28 y}{3} + 15 y = 32$

$\frac{77}{3} - \frac{77}{3} - \frac{28 y}{3} + \left(\frac{3}{3}\right) \cdot 15 y = 32 - \frac{77}{3}$

$\frac{- 28 y}{3} + \frac{45 y}{3} = \left(\frac{3}{3}\right) \cdot 32 - \frac{77}{3}$

$\frac{17 y}{3} = \frac{96}{3} - \frac{77}{3}$

$\frac{17 y}{3} = \frac{19}{3}$

$\left(\frac{3}{17}\right) \frac{17 y}{3} = \left(\frac{19}{3}\right) \left(\frac{3}{17}\right)$

$y = \frac{19}{17}$

Step 3) Substitute $\frac{22}{17}$ for $y$ in the solution to the first equation to calculate $x$:

$x = \frac{11}{3} - \left(\frac{4}{3}\right) \left(\frac{19}{17}\right)$

$x = \frac{11}{3} - \frac{76}{51}$

$x = \left(\frac{17}{17}\right) \left(\frac{11}{3}\right) - \frac{76}{51}$

$x = \frac{187}{51} - \frac{76}{51}$

$x = \frac{111}{51}$

$x = \left(\frac{3}{3}\right) \left(\frac{37}{17}\right)$

$x = \frac{37}{17}$