Solve the system of equations.
2x+8y=8
2x−4y=−16
To solve the system of equations:
1) 2x + 8y = 8
2) 2x - 4y = -16
We can solve by elimination or substitution.
Using elimination, we can multiply equation 2) by 2 to make the x-coefficients equal:
4x - 8y = -32
Then, we can add equation 1) and equation 2) together:
(2x + 8y) + (4x - 8y) = 8 + (-32)
6x = -24
Divide both sides by 6 to solve for x:
x = -24/6
x = -4
Then, we can substitute this value of x back into either equation 1) or equation 2) to solve for y. Let's use equation 1):
2(-4) + 8y = 8
-8 + 8y = 8
Add 8 to both sides:
8y = 16
Divide both sides by 8 to solve for y:
y = 2
Therefore, the solution to the system of equations is x = -4 and y = 2.
To solve this system of equations, we can use the method of elimination.
Step 1: Multiply the second equation by -2 to make the x-coefficients opposite.
-2(2x-4y) = -2(-16)
-4x + 8y = 32
Step 2: Add the two equations together to eliminate x.
(2x + 8y) + (-4x + 8y) = 8 + 32
-2x + 16y = 40
Step 3: Divide the equation by -2 to isolate y.
(-2x + 16y)/-2 = 40/-2
x - 8y = -20
So, the simplified system of equations is:
x - 8y = -20
-2x + 16y = 40
To find the value of x, we will eliminate y using a similar process.
Step 1: Multiply the first equation by 2 to make the x-coefficients opposite.
2(x - 8y) = 2(-20)
2x - 16y = -40
Step 2: Add the two equations together to eliminate y.
(-2x + 16y) + (2x - 16y) = 40 + (-40)
0 = 0
Since 0 = 0, it means that the two equations are equivalent and represent the same line. This means that there are infinitely many solutions to this system of equations.
In conclusion, the system of equations 2x + 8y = 8 and 2x - 4y = -16 has infinitely many solutions.
To solve the system of equations, we can use the method of substitution or elimination. Let's use the method of substitution here.
Step 1: Solve one equation for one variable in terms of the other variable.
Let's solve the second equation for x:
2x - 4y = -16
Add 4y to both sides:
2x = 4y - 16
Divide both sides by 2:
x = 2y - 8
Step 2: Substitute the expression for x from Step 1 into the other equation.
Now we substitute x = 2y - 8 into the first equation:
2(2y - 8) + 8y = 8
Simplify:
4y - 16 + 8y = 8
12y - 16 = 8
Step 3: Solve the resulting equation for y.
Add 16 to both sides:
12y = 24
Divide both sides by 12:
y = 2
Step 4: Substitute the value of y into either of the original equations to find the corresponding value of x.
Let's substitute y = 2 into the second equation:
2x - 4(2) = -16
Simplify:
2x - 8 = -16
Add 8 to both sides:
2x = -8
Divide both sides by 2:
x = -4
Therefore, the solution to the system of equations is x = -4 and y = 2.