y = −2x − 3
y = −2x + 2
Are the graphs of the two lines intersecting lines, the same line, or parallel lines? Explain your reasoning.
How many solutions does the system have? Explain how you can tell without solving the system.
Solve the system using any method. Show your work.
{3x − 2y = 10
−1.5x + y= 5
1st
lines are parallel ... slopes are the same ... y-intercepts are different
... no solutions
2nd
multiplying the 2nd equation by -2 gives ... 3x - 2y = -10
... same result as 1st problem
All you have to do is look at the slopes of the two lines.
the first one has slope of 3
the 2nd has a slope of -1
So they clearly must intersect.
To determine if the two lines are intersecting, parallel, or the same line, we need to compare their slopes. The given equations are in the form y = mx + b, where m represents the slope.
In the first equation, y = -2x - 3, the slope is -2.
In the second equation, y = -2x + 2, the slope is also -2.
Since both equations have the same slope, we can conclude that the lines are parallel. When lines have the same slope but different y-intercepts, they are parallel and will never intersect.
Now, let's determine the number of solutions without solving the system. Since we have parallel lines, they will never intersect, which means the system has no solution. When two lines are parallel, there is no point of intersection, so the system is considered inconsistent (no solution).
In summary, the graphs of the two lines are parallel, and the system has no solution.