Part A From the following system of equations, create a new equation by substituting the value of y from the first equation into the second equation. From the new equation, evaluate the value of x.
y = -x - 4
3x + y = 8
To substitute the value of y from the first equation into the second equation, replace y in the second equation with (-x - 4):
3x + (-x - 4) = 8
Simplifying the equation, combining like terms:
2x - 4 = 8
Adding 4 to both sides of the equation:
2x = 12
Dividing both sides by 2:
x = 6
Therefore, the value of x is 6.
Part B:
The solution to the original system of equations [is also, is not] a solution to the new equation. Solve the system of equations by SUBSTITUTION.
6y + 5x = 8
x + 3y = -7
To solve the system of equations by substitution, we'll solve one equation for one variable and substitute it into the other equation.
Let's solve the second equation for x:
x = -7 - 3y
Now, substitute this value of x into the first equation:
6y + 5(-7 - 3y) = 8
Simplifying the equation:
6y - 35 - 15y = 8
-9y - 35 = 8
-9y = 8 + 35
-9y = 43
Dividing both sides by -9:
y = -43/9
Now, substitute the value of y back into the second equation to find x:
x + 3(-43/9) = -7
Simplifying the equation:
x - 129/9 = -7
x - 129/9 = -63/9
x = -63/9 + 129/9
x = 66/9
Simplifying the fraction:
x = 22/3
Therefore, the solution to the original system of equations is x = 22/3 and y = -43/9.
Part B:
1) The solution to the original system of equations [is also, is not] a solution to the new equation.
2) Solve the system of equations by SUBSTITUTION.
6y + 5x = 8
x + 3y = -7
1) The solution to the original system of equations is also a solution to the new equation.
In the original system:
6y + 5x = 8 ------ (Equation 1)
x + 3y = -7 ------ (Equation 2)
To solve the system of equations by substitution, we'll solve Equation 2 for x and substitute it into Equation 1.
From Equation 2, we can solve for x:
x = -7 - 3y
Now, substitute this value of x into Equation 1:
6y + 5(-7 - 3y) = 8
Simplifying the equation:
6y - 35 - 15y = 8
-9y - 35 = 8
-9y = 43
Dividing both sides by -9:
y = -43/9
Now, substitute the value of y back into Equation 2 to find x:
x + 3(-43/9) = -7
Simplifying the equation:
x - 129/9 = -7
x - 129/9 = -63/9
x = -63/9 + 129/9
x = 66/9
Simplifying the fraction:
x = 22/3
Therefore, the solution to the original system of equations is x = 22/3 and y = -43/9, and this solution also satisfies the new equation.