How many solutions does the system of equations have?
y = −2x +12
4y + x −20 = 0
To solve this system of equations, we can use substitution. We know that y in the second equation is equal to -2x + 12 based on the first equation, so we can substitute that into the second equation:
4(-2x + 12) + x - 20 = 0
Simplifying this equation, we get:
-8x + 48 + x - 20 = 0
-7x + 28 = 0
-7x = -28
x = 4
Now that we know x is 4, we can substitute that back into either equation to solve for y. Let's use the first equation:
y = -2(4) + 12
y = 4
So the solution to this system of equations is (4, 4). Therefore, there is only one solution.
all you need to know is that the slopes are different.
Therefore the lines will intersect in a point -- the solution.
Yes, that is correct. If the slopes of two lines are different, then the lines intersect at exactly one point, giving us a unique solution for the system of equations.
To determine how many solutions the system of equations has, we can solve it using the elimination method or substitution method. Let's solve it using the elimination method.
First, let's rewrite the second equation in terms of y:
4y + x − 20 = 0
Rearranging the equation, we get:
x = 20 - 4y
Now we'll substitute this expression for x in the first equation:
y = -2(20 - 4y) + 12
Simplifying the equation, we get:
y = -40 + 8y + 12
Combining like terms, we have:
-7y = -28
Dividing both sides by -7, we find:
y = 4
Now we can substitute this value back into either of the original equations to find the value of x. Let's substitute it into the second equation:
4(4) + x − 20 = 0
Simplifying the equation, we get:
16 + x − 20 = 0
Rearranging the equation, we find:
x = 4
Therefore, the solution to the system of equations is x = 4 and y = 4.
Since we found a unique solution (a specific value for x and y that satisfies both equations), the system of equations has one solution.