Graph the system. Then classify it as consistent and independent, consistent and dependent, or inconsistent.
5x − y = −9
10x − 2y = −18
the 2nd equation is the 1st equation multiplied by 2
the graphs are the same line
To graph the system of equations, we need to rearrange each equation to the slope-intercept form (y = mx + b).
First equation: 5x - y = -9
Rearranging, we get: -y = -5x - 9
Dividing through by -1, we get: y = 5x + 9
Second equation: 10x - 2y = -18
Rearranging, we get: -2y = -10x - 18
Dividing through by -2, we get: y = 5x + 9
As you can see, both equations have the same slope (5) and intercept (9). This means they represent the same line.
So, when we graph the system, we get a single line:
The system is consistent and dependent because the equations represent the same line.
To graph the system of equations, we need to rearrange each equation into slope-intercept form (y = mx + b), where m represents the slope of the line and b represents the y-intercept.
For the first equation, 5x - y = -9:
1. Subtract 5x from both sides: -y = -5x - 9
2. Multiply both sides by -1 to isolate y: y = 5x + 9
Now, we can see that the slope is 5, and the y-intercept is 9. Thus, the first equation graph represents a line that passes through the point (0, 9) and has a slope of 5.
For the second equation, 10x - 2y = -18:
1. Divide the entire equation by 2 to simplify it: 5x - y = -9
2. Rearrange the equation by adding y to both sides: 5x = y - 9
3. Rearrange again by subtracting y from both sides: 5x - y = -9
The second equation is equivalent to the first equation. Therefore, both equations represent the same line when graphed. This means that the system is consistent and dependent.
To graph the system, plot the line y = 5x + 9 (representing both equations) on the coordinate plane. Since the two equations are the same, they will overlap and lie on top of each other, resulting in just a single line on the graph.
To classify the system, we determine that it is consistent and dependent because the equations are equivalent, and they have an infinite number of solutions (all points on the line). There is no unique solution for this system.