Question

In the following exercises, solve the systems of equations by elimination.

$\left\{\begin{array}{l}2x+4y=7\\ y=-\frac{1}{2}x-4\end{array}\right.$

Short answer

The given system of equations$\left\{\begin{array}{l}2x+4y=7\\ y=-\frac{1}{2}x-4\end{array}\right.$has no solution.

Step-by-step solution

Step 1. Given Information

We have given the following system of equations :-

$\left\{\begin{array}{l}2x+4y=7\\ y=-\frac{1}{2}x-4\end{array}\right.$

We have to solve this system of equations by using elimination method.

Firstly we will simplify the given system to get the system in the general form.

After that we will eliminate one variable and find the solution.

Step 2. Simplify the given system of equations

The given system of equation is :$\left\{\begin{array}{l}2x+4y=7\\ y=-\frac{1}{2}x-4\end{array}\right.$.

Now to simplify the given system take variable terms on left side in second equation, then we have :-

$\left\{\begin{array}{l}2x+4y=7\\ y+\frac{1}{2}x=-4\end{array}\right.$

Now to clear the fractions multiply both sides of second equation by $2$, then we have :-

$$\begin{gathered} \left\{\begin{array}{l}2x+4y=7\\ 2(y+\frac{1}{2}x)=2(-4)\end{array}\right. \\ =\left\{\begin{array}{l}2x+4y=7\\ 2y+x=-8\end{array}\right. \\ =\left\{\begin{array}{l}2x+4y=7\\ x+2y=-8\end{array}\right. \end{gathered}$$

Step 3. To eliminate one variable

We convert the given system of equation to the following system :-

$\left\{\begin{array}{l}2x+4y=7\\ x+2y=-8\end{array}\right.$

Now to eliminate a variable multiply the second equation by $-2$, then we have :-

role="math" $\left\{\begin{array}{l}2x+4y=7\\ -2(x+2y)=-2(-8)\end{array}\right.$

Simplify :-

$\left\{\begin{array}{l}2x+4y=7\\ -2x-4y=16\end{array}\right.$

Now add the equations :-

role="math" $0\ne 23$

This statement is false. The equations are inconsistent and so their graphs would be parallel lines.

So, the system does not have a solution.