Question

solving a system of equations using linear combination
what is the solution to this system of equations?
$4x + 5y = 7$
$3x - 2y = -12$
the solution is
$(-2, 3)$
$(3, -1)$
$(7, 3)$
$(13, -9)$
multiply each equation by a number that produces opposite coefficients for $x$ or $y$.

Explanation:

Step1: Eliminate y via scaling equations

Multiply first eq by 2: $2(4x + 5y) = 2(7) \implies 8x + 10y = 14$
Multiply second eq by 5: $5(3x - 2y) = 5(-12) \implies 15x - 10y = -60$

Step2: Add scaled equations

Add the two new equations:
$$(8x + 10y) + (15x - 10y) = 14 + (-60)$$
$$23x = -46$$

Step3: Solve for x

Isolate x:
$$x = \frac{-46}{23} = -2$$

Step4: Substitute x into first equation

Plug $x=-2$ into $4x + 5y = 7$:
$$4(-2) + 5y = 7$$
$$-8 + 5y = 7$$

Step5: Solve for y

Isolate y:
$$5y = 7 + 8 = 15$$
$$y = \frac{15}{5} = 3$$

Answer:

$(-2, 3)$