Question

in exercises 1 through 10, find all solutions of the linear systems using elimination.Then check your solutions.

4.$$\begin{gathered} \left|2x+4y=2 \\ 3x+6y=3\right| \end{gathered}$$

Short answer

The given system of equation has infinite number ofsolution

Step-by-step solution

Step 1:Transforming the equation 

To get the solution, we will transform the value of x,y.

$$\begin{gathered} \left|2x+4y=2 \\ 3x+6y=3\right| \end{gathered}$$ Into the form$$\begin{gathered} \left|x=.... \\ y=.... \\ \right| \end{gathered}$$

Eliminating the variables

In the given system of equations, we can eliminate the variables by adding or subtracting the equations.

In this system, we can eliminate the variable x from equation 1. First of all multiply the first equation by 3 and multiply the second equation by 2.

$$\begin{gathered} \left|6x+12y=6 \\ 6x+12y=6\right| \end{gathered}$$

Now subtract the first equation from the second equation.

$$\begin{gathered} \begin{array}{rcl}\left|0=0 \\ 6x+12y=6\right|& =& \left|0=0 \\ 6x+12y=6\right|\\ & & \end{array} \end{gathered}$$

After, coming the trivial solution $0=0$.Now we are left with only one solution of equation.

Given equation will form the line.

Hence, the given system of equation has infinite number of solution

Choosing the values

Let the value of x for which we will find the value of y.

$\begin{array}{rcl}x& =& 3\\ 6\left(3\right)+12y& =& 6\\ y& =& -1\\ & & \end{array}$

Checking the solution 

Now check the solution by putting the value of x and y in the given system of equation.

$$\begin{gathered} 2\left(3\right)+4(-1)=2 \\ 3\left(3\right)+6(-1)=3 \end{gathered}$$

Which is true.

Hence, the given system of the equation hasinfinite number of solutions.