How many solutions does this system have? Show your work to prove your answer.
x – 2y = 2
y = –2x + 5
To solve this system of equations, we can substitute the value of y from the second equation into the first equation.
x - 2(-2x + 5) = 2
x + 4x - 10 = 2
5x - 10 = 2
5x = 12
x = 12/5
Now, substitute the value of x back into the second equation to find the value of y.
y = -2(12/5) + 5
y = -24/5 + 25/5
y = 1/5
Therefore, the system has a unique solution, and the solution is x = 12/5, y = 1/5.
Solve the system by either substitution or elimination. Show your work to prove your answer
3x – 5y = 21
2x + y = 1
First, let's solve the second equation for y:
2x + y = 1
y = 1 - 2x
Now, substitute y = 1 - 2x into the first equation:
3x – 5(1 - 2x) = 21
3x - 5 + 10x = 21
13x - 5 = 21
13x = 26
x = 2
Now, substitute x = 2 back into the second equation to find y:
2(2) + y = 1
4 + y = 1
y = -3
Therefore, the solution to the system is x = 2, y = -3. The system has a unique solution with one solution.
What is the slope-intercept form of the equation 5x + 10y = 20? Show your work to prove your answer.
To convert the equation 5x + 10y = 20 to slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept, follow these steps:
1. Subtract 5x from both sides of the equation to isolate the term containing y:
10y = -5x + 20
2. Divide both sides by 10 to solve for y:
y = (-5/10)x + 20/10
y = (-1/2)x + 2
Therefore, the slope-intercept form of the equation 5x + 10y = 20 is y = (-1/2)x + 2.