Consider the following system of equations.
{y = 3x − 5
{y = −x + 7
a. Are the graphs of the two lines intersecting lines, the same line, or parallel lines? Explain your reasoning.
b. How many solutions does the system have? Explain how you can tell without solving the system.
I know they're intersecting lines but because I graphed out the equation and I don't think that's what the school wants me to do.
(a) they must intersect, since their slopes are different. Only parallel lines never intersect.
(b) surely you know how many times two lines can intersect. If not, look at your graph!
Oh, I see you've already done some detective work by graphing the equations! Good job, Sherlock! Let me confirm your findings in a more humorous way.
a. Are the graphs of the two lines intersecting lines, the same line, or parallel lines? Well, let's take a closer look here. The first equation, y = 3x - 5, starts with the number 3, which means it's a positive slope. The second equation, y = -x + 7, has a negative slope because of that sneaky negative sign in front of x. So, they have different slopes, meaning they're on a collision course, like two reckless cars ready to crash into each other! Hence, we can say they are intersecting lines.
b. Now, how many solutions does the system have? Ah, the mystery deepens! In this case, since the two lines intersect at a single point, there is one unique solution to the system of equations. In simpler terms, it's like finding that one missing piece that completes your jigsaw puzzle. So, without even solving the system, we can confidently say it has exactly one solution.
Keep up your detective skills, my curious friend!
To determine the intersecting nature of the two lines without graphing, we can compare their slopes.
The given system of equations is:
y = 3x - 5 ...... (Equation 1)
y = -x + 7 ...... (Equation 2)
The slopes of the lines represented by Equations 1 and 2 are 3 and -1, respectively.
a. The lines represented by the two equations have different slopes, which means they are not parallel. Since they also have different y-intercepts (-5 and 7), they cannot be the same line. Therefore, the two lines are intersecting lines.
b. When the slopes of two lines are different, there is only one point where they intersect. Therefore, the system of equations has a single solution, or a unique point of intersection.
To determine whether the given equations represent intersecting lines, same lines, or parallel lines, we can analyze the equations and the coefficients of the variables.
a. To see if the lines are intersecting, same, or parallel, we compare the slopes of the two lines. In slope-intercept form (y = mx + b), the coefficient of x represents the slope of the line.
For the equation y = 3x - 5, the slope is 3.
For the equation y = -x + 7, the slope is -1.
Since the slopes of the two lines are different (3 and -1), the lines represented by the equations are not parallel. To determine if they are intersecting or the same line, we can compare the y-intercepts.
The y-intercept of y = 3x - 5 is -5.
The y-intercept of y = -x + 7 is 7.
Since the y-intercepts of the two equations are different (-5 and 7), the lines are not the same line either.
Therefore, the given equations represent intersecting lines.
b. To determine the number of solutions without solving the system, we consider the nature of their intersection based on the slopes.
When two lines intersect, they have a unique solution.
When two lines are the same, they have infinitely many solutions.
When two lines are parallel, they have no solution.
Since we have determined that the lines are not the same and not parallel, the system of equations has a unique solution, which means it has exactly one solution.
Hence, without solving the system, we can conclude that the system has one solution based on the fact that the lines are intersecting.
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.
recall that in the form y = mx + b, m represents the slope