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 $(-2, 3)$
multiply each equation by a number that produces opposite coefficients for $x$ or $y$.

Explanation:

Step1: Eliminate y via scaling

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

Step2: Add scaled equations

Sum to cancel $10y$ and $-10y$:
$(8x + 15x) + (10y - 10y) = 14 + (-60)$
$23x = -46$

Step3: Solve for x

Divide both sides by 23:
$x = \frac{-46}{23} = -2$

Step4: Substitute x into first equation

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

Step5: Solve for y

Isolate $5y$, then divide by 5:
$5y = 7 + 8 = 15$
$y = \frac{15}{5} = 3$

Answer:

$(-2, 3)$

For the multiplication step in the interface:

• Multiply $4x + 5y = 7$ by 2
• Multiply $3x - 2y = -12$ by 5