Determine the number of solutions of this system of linear equations:%0D%0A%0D%0Ax+2y=−4%0D%0A%0D%0A+%0D%0A2%0D%0A%0D%0A=%0D%0A−%0D%0A4%0D%0A%0D%0A4x+8y=−16
To determine the number of solutions of this system of linear equations, we can look at the slopes of the two equations. If the slopes are the same, then the lines are parallel and there are no solutions. If the slopes are different, then the lines intersect at a single point and there is one solution.
First, let's rewrite the equations in slope-intercept form (y = mx + b) to determine the slopes:
1) x + 2y = -4
2y = -x - 4
y = -0.5x - 2
This equation has a slope of -0.5
2) 4x + 8y = -16
8y = -4x - 16
y = -0.5x - 2
This equation also has a slope of -0.5
Since both equations have the same slope (-0.5), the lines are parallel and there are no points of intersection. Therefore, this system of linear equations has no solutions.