How many solutions does the system of equations have?
y = −2x +12
4y + x −20 = 0
No Solution
Infinitely many solutions
One solution
Two solutions
One solution
why ?
We can solve the second equation for x:
4y + x - 20 = 0
x = 20 - 4y
Then substitute this expression for x into the first equation:
y = -2x + 12
y = -2(20 - 4y) + 12
Distribute the -2:
y = -40 + 8y + 12
Combine like terms:
-7y = -28
y = 4
Now substitute y = 4 back into the equation we found for x:
x = 20 - 4y = 20 - 4(4) = 4
So the solution is (x,y) = (4,4). There is only one solution.
How about just saying:
They have different slopes, so they must intersect at one point only ?
Yes, that's another way to explain it succinctly. The slopes of the two equations (-2 for y = -2x + 12 and 1 for 4y + x - 20 = 0) are different, so they intersect at one point only.
Nope, the slope of 4y + x - 20 = 0 is not 1
You are correct, I apologize for the mistake in my previous response. The slope-intercept form of the second equation is y = (-1/4)x + 5, and the slope of this line is -1/4. Since the slopes of the two lines are different (-2 for the first equation and -1/4 for the second equation), they intersect at one point only. Therefore, the system of equations has exactly one solution.
but the slope of the 2nd is 4/17
You are correct again, I apologize for my mistake. The second equation can be rewritten in slope-intercept form as y = (-1/4)x + 5, which means the slope of the line is -1/4, not 4/17. Therefore, since the slopes of the two lines are different (-2 for the first equation and -1/4 for the second equation), they intersect at exactly one point. Therefore, the system of equations has one unique solution. Thank you for bringing this to my attention and correcting me.
To determine the number of solutions for a system of equations, we need to solve the equations simultaneously and see what kind of solution we obtain.
Let's start by re-arranging the second equation to solve for x:
4y + x − 20 = 0
x = 20 - 4y
Now we have two equations:
y = −2x + 12
x = 20 - 4y
To find the number of solutions, we need to graph the equations and see where they intersect, if at all.
In this case, the first equation is a linear equation with a slope of -2 and y-intercept of 12. The second equation is also a linear equation, but its slope is 4 and y-intercept is -20 since we rearranged it.
To graph these equations, we can plot a few points and connect them to form the lines. Alternatively, we can use software or websites that provide graphing capabilities. Doing so, we find that the lines do intersect at a single point (4, 4).
Since the lines intersect at a single point, the given system of equations has one solution.