How many solutions does the system of equations have?
-2x=-10y+6; x+5y=3
1
2
infinitely many
none
y=4x+3; 2y-8x=3
1
2
infinitely many
none
y-7x=-14; 7y-49x=-2
1
2
infinitely many
none
Here is a quick way to find the slope of a line,
if written in the form Ax + By + C = 0
the slope of the line is -A/B
so for #1
-2x=-10y+6 ---> 2x - 10y + 6 = 0 , slope = -2/-10 = 1/5
x+5y=3 ---- slope = -1/5
so the lines have different slope, so they must intersect in 1 point.
#2
y=4x+3 in the form y = mx + b so the slope is 4
2y-8x=3 ---> 8x - 2y + 3 = 0 , slope is -2/-8 = 1/4
again, different slopes, so 1 solution
do the last one the same way.
If they have the same slope they are parallel and never cross unless they also have the same constant and are identical in which case they sit on top of each other and have an infinite number, the same y for very x
I meant to type it algebra... whoops!
So what are the number of solutions?
To determine the number of solutions, we can solve the system of equations using various methods, such as substitution, elimination, or graphing.
For the first system of equations:
-2x = -10y + 6
x + 5y = 3
We can solve this system by using the substitution method. We can rewrite the first equation as x = -5y + 3 and substitute it into the second equation:
(-5y + 3) + 5y = 3
Simplifying, we get:
-5y + 3 + 5y = 3
This simplifies to:
3 = 3
Since the equation is true (3 = 3), we have infinitely many solutions. Therefore, the correct answer for the first system of equations is "infinitely many."
For the second system of equations:
y = 4x + 3
2y - 8x = 3
We can solve this system by using the substitution method as well. We rewrite the first equation as x = (y - 3)/4 and substitute it into the second equation:
2((y - 3)/4) - 8x = 3
Simplifying, we get:
(y - 3)/2 - 8x = 3
Multiplying through by 2 to eliminate the fraction:
y - 3 - 16x = 6
Rearranging the equation:
y = 16x + 9
This is the equation of a straight line with a slope of 16. Since the two equations represent two different lines with different slopes, they intersect at a single point. Therefore, the correct answer for the second system of equations is "1."
For the third system of equations:
y - 7x = -14
7y - 49x = -2
We can solve this system by using the elimination method. First, let's multiply the first equation by 7 to make the coefficients of y in both equations equal:
7(y - 7x) = 7(-14)
7y - 49x = -98
Now we have the same coefficient for y in both equations. Subtracting the second equation from the first, we get:
7y - 7y - 49x + 49x = -98 - (-2)
0 = -96
This leads to an inconsistent equation, where 0 is not equal to -96. Therefore, there are no solutions to this system of equations. Therefore, the correct answer for the third system of equations is "none."