# Find the values for A, B, and C so that the system will have the solution (-2,4)?

## Consider the following system of equations: 3x + 4y =10 Ax + By = C How did you arrive at your answer?

Jan 10, 2018

There are infinitely many solutions; one of which is:

$A = 4 , B = - 3 , \mathmr{and} C = - 20$

#### Explanation:

You 2 equations:

$\text{1. } 3 x + 4 y = 10$
$\text{2. } A x + B y = C$

We are given that $x = - 2$ and $y = 4$; verify that the line for equation 1. contains this point:

$3 \left(- 2\right) + 4 \left(4\right) = 10$

$10 = 10 \leftarrow$ verified

We have 2 equations and 3 unknown values; this means that there are an infinite number of values for $A , B \mathmr{and} C$ and we are free to choose a solution.

I shall choose a solution so that the line for equation is 2. is perpendicular to the line for equation 1; $A = 4$ and $B = - 3$:

$4 x - 3 y = C$

To find the value of C, substitute $x = - 2$ and $y = 4$:

$4 \left(- 2\right) - 3 \left(4\right) = C$

$C = - 20$