To solve the system of equations graphically, we can plot both equations on the same coordinate system and find their point of intersection.
The equations are:
- \( y = -2x - 1 \) (Equation 1)
- \( 3x - y = -7 \) (Equation 2)
First, let's rewrite Equation 2 in slope-intercept form (y = mx + b) for easier graphing:
\[ 3x - y = -7 \] \[ -y = -3x - 7 \] \[ y = 3x + 7 \]
Now we have the two equations to graph:
- \( y = -2x - 1 \) (a line with a slope of -2 and y-intercept at (0, -1))
- \( y = 3x + 7 \) (a line with a slope of 3 and y-intercept at (0, 7))
Graphing:
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For Equation 1:
- When \( x = 0 \), \( y = -1 \) → Point (0, -1)
- When \( x = 1 \), \( y = -3 \) → Point (1, -3)
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For Equation 2:
- When \( x = 0 \), \( y = 7 \) → Point (0, 7)
- When \( x = -1 \), \( y = 4 \) → Point (-1, 4)
Now, plot these points and draw the lines for each equation. The point where the two lines intersect is the solution to the system of equations.
Intersection Point:
Using either the graph or algebra to find the intersection, we can solve for \( y \) when both \( y \) values are equal:
Set the equations equal to each other: \[ -2x - 1 = 3x + 7 \] Combine like terms: \[ -2x - 3x = 7 + 1 \] \[ -5x = 8 \] \[ x = -\frac{8}{5} = -1.6 \]
Substituting \( x = -1.6 \) into one of the equations: Using \( y = -2x - 1 \): \[ y = -2(-1.6) - 1 = 3.2 - 1 = 2.2 \]
Final Solution:
The solution to the system of equations is approximately:
\[ (-1.6, 2.2) \]
So the correct response is: Solution is (-1.6, 2.2).