Solve by substitution or elimination
1.) -6=3x-6y
4x=4+5y
2.) r+x= -1
2r-3s=6
To solve the system of equations by substitution or elimination, we'll go through each question one by one.
1.) -6 = 3x - 6y
4x = 4 + 5y
To solve this system by substitution, we'll solve one equation for one variable and substitute that expression into the other equation.
In the first equation, we have:
-6 = 3x - 6y
Rearranging this equation, we get:
3x - 6y = -6
Now, let's solve the second equation for either x or y. Since 4x is already isolated, let's solve for x in the second equation:
4x = 4 + 5y
Dividing both sides of the equation by 4, we get:
x = (4 + 5y) / 4
Now, substitute the value of x into the first equation:
3x - 6y = -6
3((4 + 5y) / 4) - 6y = -6
Simplifying this equation, we get:
3(4 + 5y) - 24y = -6
Expand and simplify further:
12 + 15y - 24y = -6
12 - 9y = -6
Now, isolate y by adding 6 to both sides:
12 - 9y + 6 = -6 + 6
18 - 9y = 0
Divide both sides of the equation by -9:
(18 - 9y) / -9 = 0 / -9
-2 + y = 0
Solving for y, add 2 to both sides:
-2 + y + 2 = 0 + 2
y = 2
Now that we have the value of y, let's substitute it back into the equation for x to find its value. Using the second equation:
4x = 4 + 5y
4x = 4 + 5(2)
4x = 4 + 10
4x = 14
Dividing both sides of the equation by 4:
x = 14 / 4
x = 3.5
Therefore, the solution for the system of equations is x = 3.5 and y = 2.
2.) r + x = -1
2r - 3s = 6
To solve this system of equations, we have two options: substitution or elimination. Let's solve it using the elimination method.
We'll start by multiplying the first equation by 2 to eliminate the x term:
2(r + x) = 2(-1)
2r + 2x = -2
Now, we can subtract this equation from the second equation:
(2r - 2r) + (2x + 3s) = (-2 - 6)
2x + 3s = -8
Now, we have a new equation:
2x + 3s = -8
To isolate one variable, we'll solve the first equation for x:
x = -1 - r
Substitute this expression for x in the second equation:
2(-1 - r) + 3s = -8
-2 - 2r + 3s = -8
Rearrange the equation to isolate s:
3s = -8 + 2 + 2r
3s = -6 + 2r
Now, divide both sides of the equation by 3 to find s:
s = (2r - 6) / 3
Now, we can substitute the expression for s back into the equation for x:
x = -1 - r
Now we have the solution for the system of equations:
x = -1 - r
s = (2r - 6) / 3
Hope this helps! Let me know if you have any further questions.