To estimate the solution to the system of linear equations given by:
- \( 2x - y = -4 \)
- \( x - y = -1 \)
we can first rewrite each equation in slope-intercept form (\(y = mx + b\)), which makes it easier to graph and find their intersection.
For the first equation \( 2x - y = -4 \):
Rearranging gives:
\[ y = 2x + 4 \]
For the second equation \( x - y = -1 \):
Rearranging gives:
\[ y = x + 1 \]
Now, let's graph the equations:
-
Graph of \( y = 2x + 4 \):
- When \( x = 0 \), \( y = 4 \) (y-intercept).
- When \( x = -2 \), \( y = 0 \) (x-intercept).
- The line will slope upwards, crossing the y-axis at 4.
-
Graph of \( y = x + 1 \):
- When \( x = 0 \), \( y = 1 \) (y-intercept).
- When \( x = -1 \), \( y = 0 \) (x-intercept).
- The line will also slope upwards, crossing the y-axis at 1.
Finding the intersection point:
The solution to the system of equations is the point where these two lines intersect.
To find this point algebraically, we can set the two equations equal to each other:
\[ 2x + 4 = x + 1 \]
Subtract \(x\) from both sides:
\[ 2x - x + 4 = 1 \]
Simplifying gives:
\[ x + 4 = 1 \]
Then we subtract 4 from both sides:
\[ x = 1 - 4 = -3 \]
Plugging \(x\) back in to find \(y\):
We can substitute \(x = -3\) into either equation; let's use \(y = x + 1\):
\[ y = -3 + 1 = -2 \]
Conclusion:
Therefore, the estimated solution to the system of equations \(2x - y = -4\) and \(x - y = -1\) is:
\[ \boxed{(-3, -2)} \]