Question

y=-2x+3
4x+2y=6 inconsistent, consistent, independent, depend

Answer 1

To determine whether the system of equations is inconsistent, consistent, independent, or dependent, we can analyze the given equations:

1. The first equation is \( y = -2x + 3 \), which is in slope-intercept form (\( y = mx + b \)). The slope (\( m \)) is \(-2\) and the y-intercept (\( b \)) is \(3\).

2. The second equation is \( 4x + 2y = 6 \). We can rearrange this equation into slope-intercept form to compare it with the first equation.

Start with:
\[
4x + 2y = 6
\]

Subtract \(4x\) from both sides:
\[
2y = -4x + 6
\]

Divide every term by \(2\):
\[
y = -2x + 3
\]

Now we can see that both equations represent the same line:
- The first equation is \( y = -2x + 3 \).
- The second equation simplifies to \( y = -2x + 3 \).

Since both equations are identical, the system of equations is **dependent**. This means there are infinitely many solutions, as every point on the line is a solution to the system.

So, the answer is: **Dependent**.