Systems of Linear Equations with No Solution - Definition & Practice

Systems with No Solution

Some systems of equations have no solutions. We know that sounds a little odd. All problems have solutions, don't they?

Nope.

The good news is that spotting a system of equations with no solution is pretty straightforward. Let's take a look at an example:

$$\begin{cases}-4x + 10y = 6\\ 2x - 5y = 3 \end{cases}$$

The first step to solve a system is ALWAYS simplify, reduce, and rearrange. We want the simplest form of each equation, and the same order of variables.

$$2x - 5y = -3$$ $$2x - 5y = 3$$

We kept the second equation the same, but divided each element in the first equation by $-2$.

Notice that both equations are in standard form.

Now that we've got our equations simplified, let's solve for one of the variables. We'll choose $x$ as our variable and $2x - 5y = -3$ as the equation.

$$2x - 5y = -3$$ $$2x = 5y -3$$ $$x = \frac{5}{2}y - \frac{3}{2}$$

Now, let's solve for $y$ just like we would normally do. Plug the value we have for $x$ into the second equation $(2x - 5y =3)$.

$$x = \frac{5}{2}y - \frac{3}{2}$$ $$2x - 5y = 3$$ $$2\Bigg(\frac{5}{2}y - \frac{3}{2}\Bigg) - 5y = 3$$ $$5y - 3 - 5y = 3$$ $$5y - 5y = 3 + 3$$ $$0 = 6$$

We know that $0$ does not equal $6$. This scenario is called nonsense. (Seriously, that's the math term for it.)

When you get a nonsense answer, you need to CHECK YOUR WORK. You must be absolutely sure that you haven't made any math errors before you decide you have nonsense. Try solving for the other variable using the other equation.

If you truly have a nonsense response, then the system of equations has no solution. Think about it like this, there is no value for $x$ or $y$ that can make $0=6$ true. There is no solution.

Graphing

Another effective way to check your work is to graph both equations in the system. You can do this in their reduced (simplified) or non-reduced (original) form—it doesn't matter.

Here are the graphs of two equations.

*Recall that the values for $x$ and $y$ come from the intersection point of the system.

Hmmmm. Our lines are parallel. That means that, no matter how far we extend them in each direction, they will never cross. There's no point of intersection. This means that there are no values for $x$ and $y$ - i.e., there is no solution.

(Video) Elimination Method

by mathman1024

This video by mathman1024 works through systems with no solution.

$\large{\text{Summary}}$

The first system he looks at is $\begin{cases}5x+2y=7\\10x+4y=-3\end{cases}$. Adding or subtracting these equations wouldn't cancel a variable. We need to multiply one of the equations before adding. We can cancel out variables if we multiply all terms in the first equation by $-2$. $$\begin{align} &-2(5x+2y)=-2(7)\\ &-10x-4y=-14 \end{align}$$ No we can eliminate. $$\begin{align} -10x-4y&=-14\\ +\ \ \ \ \ 10x+4y&=-3\\ \hline 0x+0y&=-17\\ 0&=-17 \end{align}$$ But $0=-17$ is a false statement! So what does this mean? It means that there is no solution. If we graph these lines we see that they don't intersect. In fact, they're parallel lines.

Remember that the solution of a system is the point where the lines cross. These two lines don't have an intersection, so there is no solution. Whenever the lines are parallel, there is no solution.

System of Linear Equations with No Solutions

A system has no solutions if two equations are parallel.

Let’s take a look at an example. Let's find the solution to the linear system.

$$\begin{align} 4x+2 &=y\\ 2y-8x&=10 \end{align}$$

The first step is to simplify the second equation into $y=mx+b$ form. $$\begin{align} 2y-8x&=10\\ y-4x&=5\\ y&=4x+5 \end{align}$$

Now, the two equations are: $$\begin{align} y=4x+2\\ y=4x+5 \end{align}$$

The coefficients of the variables are the same, so these lines have the same slope. They will never overlap since they have different $y$-intercepts. We can graph the two equations to visualize this:

The red line is the graph of $y=4x+2$, and the blue line is the graph of $y=4x+5$.

When two equations have the same slope but different $y$-axis, they are parallel. Since there are no intersection points, the system has no solutions.

Now let’s try an example.

The first equation has a common factor of $2$, so it can be reduced by dividing everything by $2$. Thus, our new system of equations is: $$\begin{align} 3x+y&=5\\ y+9&=-3x \end{align}$$ If we solve for $y$ in both equations, we have: $$y=-3x+5$$ $$y=-3x-9$$ Since the two equations have the same slope but different $y$-intercept, they are parallel. The two lines never intersect, so there is no solution.

You can also solve the system via substitution or elimination.