Consider the following system of equations.
{ y=3x−5
{ y=−x+7
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
Since the slope are NOT the same and the first one is positive (3) that means the graph slopes UPWARDS when looked at from left to right.
While the second line has a slope of -1 so the graph slopes DOWNWARDS when looked at from left to right.
Thus the lines intersect in ONE place : )
You can sketch them to find the point of intersection if you wish.
Well, the graph of the first equation, y = 3x - 5, has a positive slope of 3, which means it goes uphill. On the other hand, the second equation, y = -x + 7, has a negative slope of -1, which means it goes downhill.
Since the slopes are different, the two lines are not parallel. And since they have different y-intercepts, they are not the same line either. So, that can only mean one thing...
They are intersecting lines! It's like two people on different roller coasters passing each other – they intersect at some point.
As for the solutions, since the lines intersect, there is one solution to the system of equations. You can tell this without solving the system because the lines actually cross each other at a single point. It's like having a burger with exactly one pickle – you don't need to eat it to know there's only one.
To determine whether the graphs of the two lines intersecting lines, the same line, or parallel lines, we need to compare their slopes.
The given system can be written in slope-intercept form as:
Line 1: y = 3x - 5
Line 2: y = -x + 7
Comparing the coefficients of x in both equations, we see that the slope of Line 1 is 3 and the slope of Line 2 is -1.
When the slopes of two lines are different, the lines will intersect at a single point. Therefore, we can conclude that the lines represented by the given equations are intersecting lines.
Now, to determine the number of solutions that the system has, we rely on the fact that intersecting lines have exactly one point of intersection.
Therefore, the system of equations has a single solution.
To determine whether the given system of equations represents intersecting lines, the same line, or parallel lines, we can compare their slopes.
Let us rewrite the equations in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
Equation 1: y = 3x - 5
The slope of Equation 1 is 3.
Equation 2: y = -x + 7
The slope of Equation 2 is -1.
If the slopes of the two equations are equal, the lines are the same. If the slopes are different, the lines are intersecting. If the slopes are equal and the y-intercepts differ, the lines are parallel.
In this case, since the slopes of the two equations are different (3 and -1), the lines represented by these equations are intersecting lines.
To determine the number of solutions, we use the fact that intersecting lines have exactly one solution. So, the given system has only one solution.
Therefore, the graphs of the two lines are intersecting lines and the system has one solution.
By the "Just Look at it Theorem" the lines have different slopes,
so what does that tell you about the lines ?