Y = 2 + 3x
4x - 3y = -16
To solve this system of equations, we can use substitution or elimination method.
Substitution method:
We are given that Y = 2 + 3x.
Substituting this value of Y into the second equation, we get:
4x - 3(2 + 3x) = -16
4x - 6 - 9x = -16
-5x - 6 = -16
-5x = -10
x = 2
Substituting the value of x into the equation Y = 2 + 3x, we get:
Y = 2 + 3(2)
Y = 2 + 6
Y = 8
Therefore, the solution to the system of equations is x = 2 and y = 8.
Alternatively, we can also solve this system of equations using the elimination method:
Multiply the first equation through by 3 to make the coefficients of y in both equations the same:
3Y = 6 + 9x
Now we have the system of equations:
3Y = 6 + 9x
4x - 3y = -16
Add the two equations together to eliminate y:
3Y + (4x - 3Y) = (6 + 9x) + (-16)
4x - 3y = -10 + 9x
4x - 3y - 9x = -10
-5x - 3y = -10
Divide through by -1 to get the coefficient of x positive:
5x + 3y = 10
Now we have the system of equations:
-5x - 3y = -10
5x + 3y = 10
Add the two equations together to eliminate x:
(-5x - 3y) + (5x + 3y) = (-10) + 10
0 = 0
The equation becomes an identity, which means that it is true for all values of x and y. This indicates that the two equations are actually the same line and therefore have infinitely many solutions.
So, the system of equations is consistent and dependent, and the solution is any point on the line represented by the equation. The graph of the line would be a straight line passing through all points (x, y) that satisfy the given equation.
To solve the equations Y = 2 + 3x and 4x - 3y = -16, we can use the method of substitution.
First, we isolate one of the variables in either equation. Let's isolate Y in the first equation.
Step 1: Start with the equation Y = 2 + 3x.
Subtract 2 from both sides to isolate Y:
Y - 2 = 2 + 3x - 2
Y - 2 = 3x
Step 2: Now, we have Y = 3x + 2.
Next, substitute this expression for Y in the second equation.
Step 3: Replace Y in the equation 4x - 3y = -16 with 3x + 2:
4x - 3y = -16
4x - 3(3x + 2) = -16
Step 4: Distribute the -3 to the terms inside the parentheses:
4x - 3y = -16
4x - 9x - 6 = -16
Step 5: Combine like terms on the left side of the equation:
4x - 9x - 6 = -16
-5x - 6 = -16
Step 6: Add 6 to both sides of the equation to isolate the variable:
-5x - 6 + 6 = -16 + 6
-5x = -10
Step 7: Divide both sides of the equation by -5 to solve for x:
-5x / -5 = -10 / -5
x = 2
Step 8: Substitute the value of x into either of the original equations to solve for y.
Let's use the equation Y = 3x + 2:
Y = 3(2) + 2
Y = 6 + 2
Y = 8
Therefore, the solution to the system of equations Y = 2 + 3x and 4x - 3y = -16 is x = 2 and y = 8.