# How do you determine whether a linear system has one solution, many solutions, or no solution when given 2x+5y=-16 and 6x+y=20?

Oct 20, 2015

A linear system has one solution when the two lines comprising the system intersect once. A linear system has many (infinite) solutions when the two lines are the same (such as $y = x + 3$ and $2 y = 2 x + 6$). And a linear system has no solution when the lines never intersect (in other words, they're parallel; their slopes are equal).

#### Explanation:

Here we have the system
$2 x + 5 y = - 16$
$6 x + y = 20$

To determine the nature of the solutions (whether we have 0, 1, or infinitely many), we put each equation into the form $y = m x + b$, where $m$ is the slope and $b$ is the y-intercept. To do this, we simply solve for $y$. Let's start with the first equation:

$2 x + 5 y = - 16$
$5 y = - 16 - 2 x$, or equivalently, $5 y = - 2 x - 16$ (subtracting 2x from both sides)
$y = \frac{- 2 x - 16}{5}$ (dividing both sides by 5)
$y = - \frac{2 x}{5} - \frac{16}{5}$ (putting it into the $y = m x + b$ format)

We see that our slope, $m$, is $- \frac{2}{5}$ and our y-intercept, $b$, is $- \frac{16}{5}$. Now onto the next equation:

$6 x + y = 20$
$y = 20 - 6 x$, or equivalently, $y = - 6 x + 20$ (subtracting 6x from both sides)

This time, our slope is $- 6$ and our y-int. is $20$. Since we have two different slopes (-6 and -2/5), that tells us our linear system has 1 solution. If the slopes were equal, we would have 0 solutions, and if the lines themselves (both slope and y-int.) were the same, we would have infinite solutions. To prove we only have one solution, look at the graph below and see how the lines intersect once.

Graph from desmos.com/calculator.