If 2x + y = 23 and 4x - y = 19, find the values of x - 3y and 5y - 2x.

Answer

To solve the simultaneous equations \(2x + y = 23\) and \(4x - y = 19\), and then find the values of \(x - 3y\) and \(5y - 2x\), we will follow these steps:

### Step 1: Write down the equations
We have the following two equations:
1. \(2x + y = 23\) (Equation 1)
2. \(4x - y = 19\) (Equation 2)

### Step 2: Add the two equations
To eliminate \(y\), we can add both equations:
\[ (2x + y) + (4x - y) = 23 + 19 \]
This simplifies to:
\[ 2x + 4x + y - y = 42 \]
\[ 6x = 42 \]

### Step 3: Solve for \(x\)
Now, divide both sides by 6:
\[ x = \frac{42}{6} = 7 \]

### Step 4: Substitute \(x\) back into one of the equations to find \(y\)
We can substitute \(x = 7\) into Equation 1:
\[ 2(7) + y = 23 \]
This simplifies to:
\[ 14 + y = 23 \]
Now, solve for \(y\):
\[ y = 23 - 14 = 9 \]

### Step 5: Find the values of \(x - 3y\) and \(5y - 2x\)
Now that we have \(x = 7\) and \(y = 9\), we can find the required expressions.

1. For \(x - 3y\):
\[ x - 3y = 7 - 3(9) = 7 - 27 = -20 \]

2. For \(5y - 2x\):
\[ 5y - 2x = 5(9) - 2(7) = 45 - 14 = 31 \]

### Final Answers
Thus, the values are:
- \(x - 3y = -20\)
- \(5y - 2x = 31\)

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