Consider the system of linear equations below
-3x + 4y = 24
6x + 2y = -18
Determine how to rewrite one of the two equations above in the form ax + by = c, where a, b, and c are constants so that the sum of the new equation and the unchanged equation from the original system results in an equation of one variable.
To rewrite one of the equations in the form ax + by = c, we can multiply the equation by a constant so that the coefficients of x and y are the same as the coefficients in the other equation.
Let's rewrite the first equation -3x + 4y = 24.
To make the coefficient of x the same as the second equation (6x), we can multiply the entire equation by -2:
-2(-3x + 4y) = -2(24)
6x - 8y = -48
Now, if we add this new equation to the unchanged equation from the original system (6x + 2y = -18), we get:
(6x - 8y) + (6x + 2y) = -48 + (-18)
12x - 6y = -66
Therefore, the sum of the new equation (6x - 8y = -48) and the unchanged equation (6x + 2y = -18) results in the equation:
12x - 6y = -66, an equation of one variable.