The point of intersection of the graphs of the equations of the system
Ax – 4y = 9
4x + By = –1
is (–1, –3). Explain how to find the values of A and B, then find these values.
The point (-1,-3) must lie on both curves. Thus
-A -4*(-3) = 9
-A +12 = 9
A = 3
and
4*(-1) +B*(-3) = -1
Solve that for B
x = - 1
y = - 3
A x – 4 y = 9
A * ( - 1 ) - 4 * ( - 3 ) = 9
- A + 12 = 9
- A = 9 - 12
- A = -3 Multiply both sides by - 1
A = 3
4 x + B y = –1
4 * ( - 1 ) + B * ( - 3 ) = - 1
- 4 - 3 B = - 1
- 3 B = - 1 + 4
- 3 B = 3 Divide both sides by - 3
B = 3 / - 3
B = - 1
To find the values of A and B in the given system of equations, you can use the fact that the point of intersection of the graphs lies on both lines simultaneously.
To find A:
1. Substitute the x and y values of the given point, (-1, -3), into the first equation (Ax - 4y = 9).
2. Replace x with -1 and y with -3 in the equation: A(-1) - 4(-3) = 9.
3. Simplify the equation: -A + 12 = 9.
4. Solve for A: -A = 9 - 12 = -3. Therefore, A = 3.
To find B:
1. Substitute the x and y values of the given point, (-1, -3), into the second equation (4x + By = -1).
2. Replace x with -1 and y with -3 in the equation: 4(-1) + B(-3) = -1.
3. Simplify the equation: -4 - 3B = -1.
4. Solve for B: -3B = -1 + 4 = 3. Therefore, B = -1.
Therefore, the values of A and B in the given system of equations are A = 3 and B = -1.