# How do you solve 3x+y=5 and x-2y=11 using substitution?

Jan 18, 2017

See the entire solution process below:

#### Explanation:

Step 1) Solve the first equation for $y$:

$3 x - \textcolor{red}{3 x} + y = 5 - \textcolor{red}{3 x}$

$0 + y = 5 - 3 x$

$y = 5 - 3 x$

Step 2) Substitute $\textcolor{red}{5 - 3 x}$ for $y$ in the second equation and solve for $x$:

$x - 2 \left(\textcolor{red}{5 - 3 x}\right) = 11$

$x - \left(2 \times \textcolor{red}{5}\right) + \left(2 \times \textcolor{red}{3 x}\right) = 11$

$x - 10 + 6 x = 11$

$7 x - 10 = 11$

$7 x - 10 + \textcolor{red}{10} = 11 + \textcolor{red}{10}$

$7 x - 0 = 21$

$7 x = 21$

$\frac{7 x}{\textcolor{red}{7}} = \frac{21}{\textcolor{red}{7}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{7}}} x}{\cancel{\textcolor{red}{7}}} = 3$

$x = 3$

Step 3) Substitute $\textcolor{red}{3}$ for $x$ in the solution to the first equation at the end of Step 1

$y = 5 - \left(3 \times \textcolor{red}{3}\right)$

$y = 5 - 9$

$y = - 4$

The solution is $x = 3$ and $y = - 4$

Jan 18, 2017

$x = - \frac{1}{7}$, $y = - \frac{39}{7}$. Explanation below.

#### Explanation:

Using the second equation, we can express $y$ in terms of $x$ as such:

$- 2 y = 11 - x \implies y = - \frac{11}{2} + \frac{x}{2}$.

Substituting this $y$ value in the first equation:

$3 x + \left(- \frac{11}{2} + \frac{x}{2}\right) = 5 \implies \frac{7 x}{2} = - \frac{1}{2} \implies$

$\implies 7 x = - 1 \implies x = - \frac{1}{7}$.

Now, we can substitute this $x$ value in the second equation to find $y$ (any equation will do, but $x$ is easier to evaluate than $3 x$)

$- \frac{1}{7} - 2 y = 11 \implies 2 y = - \frac{78}{7} \implies y = - \frac{78}{14} = - \frac{39}{7}$