To solve the system of equations by graphing, we first need to rewrite the equations in slope-intercept form (y = mx + b) for easier graphing.
The given equations are:
- \( x + 2y = -6 \)
- \( x + y = -1 \)
Step 1: Rearranging into slope-intercept form
For the first equation: \[ x + 2y = -6 \] Subtract \( x \) from both sides: \[ 2y = -x - 6 \] Now divide by 2: \[ y = -\frac{1}{2}x - 3 \]
For the second equation: \[ x + y = -1 \] Subtract \( x \) from both sides: \[ y = -x - 1 \]
Step 2: Graph the equations
Now we have the two equations in slope-intercept form:
- \( y = -\frac{1}{2}x - 3 \)
- \( y = -x - 1 \)
Graphing the first equation \( y = -\frac{1}{2}x - 3 \):
- The y-intercept is -3 (the point (0, -3)).
- The slope is -1/2, which means for each 2 units you move right (positive x), you move 1 unit down (negative y).
Graphing the second equation \( y = -x - 1 \):
- The y-intercept is -1 (the point (0, -1)).
- The slope is -1, which means for each 1 unit you move right, you move 1 unit down.
Step 3: Finding the intersection point
To find the solution to the system of equations, we need to find the point where the lines intersect.
From graphing, we can identify the intersection point. Alternatively, we can set the equations equal to each other to find this point algebraically.
Setting the two equations equal: \[ -\frac{1}{2}x - 3 = -x - 1 \]
Multiplying through by -2 to eliminate the fraction gives: \[ x + 6 = 2x + 2 \] Now, rearranging gives: \[ 6 - 2 = 2x - x \] \[ 4 = x \]
Now substituting \( x = 4 \) back into one of the original equations to find \( y \): Using \( x + y = -1 \): \[ 4 + y = -1 \] Subtracting 4 from both sides gives: \[ y = -5 \]
Conclusion
The solution to the system of equations is: \[ \boxed{(4, -5)} \]
This is the point where the two lines intersect on the graph.