The graphs of 5x-3y=35, 7x-3y=43, and 4x-ay=61 all intersect at the same point. Find the value of "a".
Do I solve the systems and then plug in? I am not sure haha.
Here is how I would have expected my students to show the steps:
subtract the first equation from the second :
7x-3y=43
5x - 3y = 35
2x = 8
x = 4
sub that back into 5x-3y=35
20-3y=35
-3y = 15
y = -5
So the first two lines intersect at (4,-5)
But the third line of 4x-ay=61 also runs through that point, thus
4(4) - a(-5) = 61
16 + 5a = 61
5a = 45
a = 9
5x-3y=35 7x-3y=43 solve simultaneously by subtracting 1 from 2:2x=8 x=8/2=4 5(4)-3y=35 -3y=35-20 y=15/-3=-5 since they intersect,x and y are similar:4(4)-a(-5)=61 5a=61-16 5a=45 a=9
Yes, to find the value of "a" such that the three given equations intersect at the same point, we can solve the system of equations to find the x-coordinate and y-coordinate of that point. Once we have the coordinates, we can then substitute them into the third equation and solve for "a".
Let's start by solving the system of equations. We have the following equations:
1) 5x - 3y = 35
2) 7x - 3y = 43
3) 4x - ay = 61
To solve this system, we can use the method of elimination. The goal is to eliminate one of the variables so that we can solve for the other variable.
First, let's eliminate the "y" variable. To do this, we can take equation 2 and subtract equation 1:
(7x - 3y) - (5x - 3y) = 43 - 35
2x = 8
x = 4
Now that we have the x-coordinate of the point of intersection, we can substitute it back into either equation 1 or 2 to find the y-coordinate.
Let's substitute x = 4 into equation 1:
5(4) - 3y = 35
20 - 3y = 35
-3y = 15
y = -5
So the coordinates of the point of intersection are (4, -5).
Now, we substitute these values, x = 4 and y = -5, into equation 3 to solve for "a":
4(4) - a(-5) = 61
16 + 5a = 61
5a = 61 - 16
5a = 45
a = 9
Therefore, the value of "a" that makes the three given equations intersect at the same point is 9.