Solve by using the subatitution method. (Please keep fractions as they are, don't convert to decimal.)
1. y= -3x - 1
2x - 3y= -8
2. 1 + 3y= 10
5x + 2y= 6
3. 2x - 6y= -2
x= 3y - 1
4. y= 1/7x + 3
x - 7y= -4
I will do #3, you try the others
3.
2x - 6y = -2
x = 3y-1
sub into the 1st:
2(3y-1) - 6y = -2
6y - 2 - 6y = -2
0 = 0
Ahhh, since the variables dropped out and we ended up with a TRUE statement, the two equations are really one and the same equation.
(if you simplify the first, you get the second ...
2x - 6y = -2
divide by 2
x - 3y = -1
x = 3y - 1, which was the second equation )
So any ordered pair which satisfies the 1st will obviously also satisfy the second.
Had we ended up with a false statement, such as 3 = 0, there would have been no solution at all
Hm... so, 0=0 is the only answer for #3? You don't have to do anything else? Like plugging in for the second equation or something.
To solve these systems of equations using the substitution method, follow these steps:
1. Start with one of the equations and solve for one variable in terms of the other.
2. Substitute the expression you found in step 1 into the other equation.
3. Solve the resulting equation for the remaining variable.
4. Substitute the value found in step 3 back into one of the original equations to solve for the other variable.
5. Write the solution as an ordered pair (x, y).
Let's solve the given systems of equations using this method:
1. y = -3x - 1 ...(Equation 1)
2x - 3y = -8 ...(Equation 2)
From Equation 1, we can express y in terms of x:
y = -3x - 1
Now, we substitute this expression for y into Equation 2:
2x - 3(-3x - 1) = -8
Simplify the equation:
2x + 9x + 3 = -8
11x + 3 = -8
11x = -8 - 3
11x = -11
x = -11/11
x = -1
Substitute x = -1 back into Equation 1 to solve for y:
y = -3(-1) - 1
y = 3 - 1
y = 2
The solution to the system of equations is (x, y) = (-1, 2).
Now, let's solve the remaining systems:
2. 1 + 3y = 10 ...(Equation 1)
5x + 2y = 6 ...(Equation 2)
From Equation 1, we can express y in terms of x:
1 + 3y = 10
3y = 10 - 1
3y = 9
y = 9/3
y = 3
Substitute y = 3 back into Equation 2:
5x + 2(3) = 6
5x + 6 = 6
5x = 6 - 6
5x = 0
x = 0/5
x = 0
The solution to the system of equations is (x, y) = (0, 3).
3. 2x - 6y = -2 ...(Equation 1)
x = 3y - 1 ...(Equation 2)
Substitute x = 3y - 1 into Equation 1:
2(3y - 1) - 6y = -2
6y - 2 - 6y = -2
-2 = -2 (this equation is true)
Since the equation -2 = -2 is true, the two equations represent the same line, and there are infinitely many solutions.
4. y = 1/7x + 3 ...(Equation 1)
x - 7y = -4 ...(Equation 2)
From Equation 1, we can express y in terms of x:
y = 1/7x + 3
Substitute this expression for y into Equation 2:
x - 7(1/7x + 3) = -4
Simplify the equation:
x - x - 21 = -4
-21 = -4 (this equation is false)
Since the equation -21 = -4 is false, the two equations are inconsistent and have no solution.
In summary:
1. The solution to the first system is (-1, 2).
2. The solution to the second system is (0, 3).
3. The third system has infinitely many solutions.
4. The fourth system has no solution.