Question

How many solutions to this system of linear equations
2x+y=6
4x+2y=8

Answer 1

What is the solution to the equations in slope intercept form?

Answer 2

To determine the number of solutions to the given system of linear equations:

1. \(2x + y = 6\) (Equation 1)
2. \(4x + 2y = 8\) (Equation 2)

First, let's check if the second equation can be simplified. Observe that:

\[
4x + 2y = 8
\]

Dividing the entire equation by 2 gives:

\[
2x + y = 4
\]

Now, we have the following two equations:

1. \(2x + y = 6\)
2. \(2x + y = 4\)

Next, we can see that both equations are in the form \(2x + y = k\), with \(k\) being different for each equation (6 and 4 respectively). This means that the two lines represented by these equations are parallel lines and do not intersect.

Since the lines are parallel, they have no solutions in common.

Therefore, the system of linear equations has **no solutions**.

Answer 3

To write the given equations in slope-intercept form, we need to express them as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

1. **First equation:**
\[
2x + y = 6
\]
To isolate \(y\), we can rearrange the equation:
\[
y = -2x + 6
\]

Here, the slope \(m\) is \(-2\) and the y-intercept \(b\) is \(6\).

2. **Second equation:**
\[
4x + 2y = 8
\]
First, divide the entire equation by 2 to simplify:
\[
2x + y = 4
\]
Now, rearranging it to get \(y\) alone:
\[
y = -2x + 4
\]

Here, the slope \(m\) is also \(-2\) and the y-intercept \(b\) is \(4\).

### Summary:
- The first equation in slope-intercept form is \(y = -2x + 6\).
- The second equation in slope-intercept form is \(y = -2x + 4\).

These two lines have the same slope \(-2\) but different y-intercepts (\(6\) and \(4\)), confirming that they are parallel and thus have no solutions in common.