The Problem

We are given the following system of equations:

1. $3x + 2y = 5$

2. $2x – 3y = 7$

Step 1: Create Matching Coefficients

To eliminate $x$, we need the coefficients of $x$ in both equations to be the same. We can achieve this by finding the least common multiple of 3 and 2, which is 6.

• Multiply Equation (1) by 2: $6x + 4y = 10$

• Multiply Equation (2) by 3: $6x – 9y = 21$

Step 2: Eliminate $x$

Now, we subtract the second resulting equation from the first to “eliminate” the $x$ term:

This simplifies to:

Step 3: Solve for $x$

Now that we have the value for $y$, substitute it back into Equation (1) to find $x$:

Add $\frac{22}{13}$ to both sides:

Divide by 3 to isolate $x$:

The Final Result

The solution to the system of equations is the coordinate where these two lines intersect:

Note: There is a small typo in the final line of the original image (it lists the denominator of $y$ as 3 instead of 13), but the steps above show the mathematically correct derivation!