Answer
The solution is $(2, 1)$.
Work Step by Step
We first need to modify the two equations so that the coefficients of one of the variables differs only in sign. With this modification, we will be able to cancel out one of the variables and solve for the other one:
Let us modify the equations first. We see that if we multiply the first equation by $2$ and the second equation by $-3$, the coefficient of the $x$ term will be $6$ and $-6$. The coefficients will now only differ in sign, so we can cancel them out when we add the two equations. Let us do the multiplication:
$2(3x + y) = 2(7)$
$-3(2x - 5y) = -3(-1)$
Use distributive property:
$2(3x) + 2(y) = 2(7) $
$-3(2x) - (3)(-5y) = -3(-1)$
Let's multiply out the terms:
$6x + 2y = 14$
$-6x + 15y = 3$
We can now cancel out the $x$ terms to get:
$2y = 14$
$15y = 3$
Now we add both sides of the two equations to get:
$17y = 17$
Divide both sides of the equation by $17$ to solve for $y$:
$y = 1$
Now that we have the value for $y$, we can plug this value into one of the equations to solve for $x$. Let's use the first equation:
$3x + 1 = 7$
Subtract $1$ from both sides of the equation to isolate the variable on one side of the equation and the constants on the other:
$3x = 6$
Divide each side of the equation by $3$ to solve for $x$:
$x = 2$
The solution is $(2, 1)$.
