Question

equations $-2x + 3y = -5$ and $3x - 4y = 6$.

Explanation:

Assuming the problem is to solve the system of linear equations \(-2x + 3y = -5\) and \(3x - 4y = 6\) (since the original problem statement seems incomplete but this is a common problem with these equations), we can use the elimination method.

Step 1: Eliminate one variable

First, we'll make the coefficients of \(x\) (or \(y\)) the same in both equations. Let's eliminate \(x\). Multiply the first equation by \(3\) and the second equation by \(2\):

For the first equation \(-2x + 3y = -5\), multiplying by \(3\) gives:
\(3\times(-2x + 3y)=3\times(-5)\)
\(-6x + 9y = -15\) --- (1)

For the second equation \(3x - 4y = 6\), multiplying by \(2\) gives:
\(2\times(3x - 4y)=2\times6\)
\(6x - 8y = 12\) --- (2)

Step 2: Add the two new equations

Now, add equation (1) and equation (2) to eliminate \(x\):

\((-6x + 9y)+(6x - 8y)=-15 + 12\)

Simplify the left - hand side: \(-6x+6x + 9y-8y=y\)

Simplify the right - hand side: \(-3\)

So, we get \(y=-3\)

Step 3: Substitute \(y = - 3\) into one of the original equations

Substitute \(y = - 3\) into the first original equation \(-2x+3y=-5\):

\(-2x+3\times(-3)=-5\)

\(-2x-9=-5\)

Step 4: Solve for \(x\)

Add \(9\) to both sides of the equation:

\(-2x-9 + 9=-5 + 9\)

\(-2x=4\)

Divide both sides by \(-2\):

\(x=\frac{4}{-2}=-2\)

Answer:

The solution to the system of equations is \(x=-2\) and \(y = - 3\)