Question

1. $4x - 9y = 17$

$2x + 18y = -14$

1. $-4x - 9y = 5$

$-10x - 18y = -10$

Explanation:

Problem 14: Solve the system \(

$$\begin{cases}4x - 9y = 17\\2x + 18y = -14\end{cases}$$

\)

Step 1: Eliminate \(x\) by multiplying the second equation

Multiply the second equation \(2x + 18y = -14\) by \(2\) to make the coefficient of \(x\) equal to \(4\) (opposite sign to the first equation's \(x\) coefficient? Wait, no, first equation has \(4x\), second after multiplying by \(2\) will have \(4x\). Wait, actually, to eliminate \(x\), we can multiply the second equation by \(2\) to get \(4x + 36y = -28\), then subtract the first equation from it? Wait, no, let's do it properly.

Wait, first equation: \(4x - 9y = 17\)

Second equation: \(2x + 18y = -14\). Let's multiply the second equation by \(2\): \(4x + 36y = -28\)

Now, subtract the first equation from this new equation:

\((4x + 36y) - (4x - 9y) = -28 - 17\)

Simplify left side: \(4x + 36y - 4x + 9y = 45y\)

Right side: \(-45\)

So \(45y = -45\)

Step 2: Solve for \(y\)

Divide both sides by \(45\): \(y = \frac{-45}{45} = -1\)

Step 3: Substitute \(y = -1\) into the first equation to find \(x\)

First equation: \(4x - 9(-1) = 17\)

Simplify: \(4x + 9 = 17\)

Subtract \(9\) from both sides: \(4x = 17 - 9 = 8\)

Divide by \(4\): \(x = \frac{8}{4} = 2\)

Step 1: Eliminate \(y\) by multiplying the first equation by \(2\)

First equation: \(-4x - 9y = 5\). Multiply by \(2\): \(-8x - 18y = 10\)

Second equation: \(-10x - 18y = -10\)

Now, subtract the second equation from the new first equation:

\((-8x - 18y) - (-10x - 18y) = 10 - (-10)\)

Simplify left side: \(-8x - 18y + 10x + 18y = 2x\)

Right side: \(20\)

So \(2x = 20\)

Step 2: Solve for \(x\)

Divide both sides by \(2\): \(x = \frac{20}{2} = 10\)

Step 3: Substitute \(x = 10\) into the first equation to find \(y\)

First equation: \(-4(10) - 9y = 5\)

Simplify: \(-40 - 9y = 5\)

Add \(40\) to both sides: \(-9y = 5 + 40 = 45\)

Divide by \(-9\): \(y = \frac{45}{-9} = -5\)

Answer:

\(x = 2\), \(y = -1\)

Problem 16: Solve the system \(

$$\begin{cases}-4x - 9y = 5\\-10x - 18y = -10\end{cases}$$

\)