Name the point of intersection for the following system, if it exists.
4x – 3y = -3
3x – y = -6
3 x – y = - 6
3 x + 6 = y
y = 3 x + 6
4 x – 3 y = - 3
4 x - 3 ( 3 x + 6 ) = - 3
4 x - 3 * 3 x - 3 * 6 = - 3
4 x - 9 x - 18 = - 3
- 5 x - 18 = - 3
- 5 x = - 3 + 18
- 5 x = 15 Divide both sides by - 5
x = 15 / - 5
x = - 3
y = 3 x + 6
y = 3 * ( - 3 ) + 6
y = - 9 + 6
y = - 3
Point with coordinates:
x = - 3
y = - 3
To find the point of intersection for the given system of equations, we need to solve the equations simultaneously. Here's a step-by-step explanation of the process:
1. Start by writing the system of equations:
4x – 3y = -3 ---(1)
3x – y = -6 ---(2)
2. We can use the method of substitution or elimination to solve the system. Let's use the elimination method.
3. Multiply equation (2) by 3 to make the coefficients of x in both equations the same:
9x - 3y = -18 ---(3)
4. Now, we can subtract equation (1) from equation (3) to eliminate the y variable:
(9x - 3y) - (4x - 3y) = -18 - (-3)
9x - 3y - 4x + 3y = -18 + 3
5x = -15
5. Simplify equation (4) to solve for x:
5x = -15
Divide both sides by 5:
x = -3
6. Now substitute the value of x back into either equation (1) or (2) to solve for y. Let's use equation (1):
4x - 3y = -3
4(-3) - 3y = -3
-12 - 3y = -3
-3y = -3 + 12
-3y = 9
7. Simplify equation (6) to solve for y:
-3y = 9
Divide both sides by -3:
y = -3
8. Therefore, the solution to the system of equations is x = -3 and y = -3.
So, the point of intersection for the given system is (-3, -3).