Find the values of x and y for the system: $7x - 15y = 2$ and x + $2y = 3$

Step 1: Express one variable in terms of the other

From the second equation, $x + 2y = 3$, we can easily express $x$ in terms of $y$. This is a common first step in the substitution method, where we isolate one variable to substitute it into the other equation. We achieve this by subtracting $2y$ from both sides of the equation.

Step 2: Substitute the expression into the other equation

Now, we substitute the expression for $x$, which is $3 - 2y$, into the first equation, $7x - 15y = 2$. This substitution will result in a linear equation that contains only the variable $y$, making it solvable.

Step 3: Simplify and solve for y

We first distribute the 7 into the parenthesis, which gives us $21 - 14y$. Then, we combine the like terms, $-14y$ and $-15y$, to get $-29y$. Next, we subtract 21 from both sides of the equation and finally divide by $-29$ to find the value of $y$.

Step 4: Substitute the value of y back into the expression for x

Now that we have the value of $y = \frac{19}{29}$, we substitute it back into the expression for $x$ that we derived in the first step: $x = 3 - 2y$. We multiply 2 by $\frac{19}{29}$ and then subtract the resulting fraction from 3, finding a common denominator to perform the subtraction.

Step 5: Verify the solution

To ensure our solution is correct, we substitute the calculated values of $x = \frac{49}{29}$ and $y = \frac{19}{29}$ into both original equations. If both equations hold true, our solution is verified. In this case, both equations are satisfied.

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