Find the solution to the following systems of equations: a) { x - 6y = 17, 5x + 6y = 13; б) { 4x - 7y = -12, -4x + 3y = 12; в) { 3x + 2y = 5, -5x + 2y = 45; г) { 9x - 4y = -13, 9x - 2y = -20.

The task is to solve four systems of linear equations. We will solve each system separately.

System a)

The system of equations is:

• $$x - 6y = 17$$
• $$5x + 6y = 13$$

Step 1: Add the two equations together to eliminate y.

$$(x - 6y) + (5x + 6y) = 17 + 13$$

$$6x = 30$$

Step 2: Solve for x.

$$x = \frac{30}{6} = 5$$

Step 3: Substitute the value of x into the first equation to solve for y.

$$5 - 6y = 17$$

$$-6y = 17 - 5$$

$$-6y = 12$$

$$y = \frac{12}{-6} = -2$$

Solution for a): $$x = 5, y = -2$$

System б)

The system of equations is:

• $$4x - 7y = -12$$
• $$-4x + 3y = 12$$

Step 1: Add the two equations together to eliminate x.

$$(4x - 7y) + (-4x + 3y) = -12 + 12$$

$$-4y = 0$$

Step 2: Solve for y.

$$y = \frac{0}{-4} = 0$$

Step 3: Substitute the value of y into the first equation to solve for x.

$$4x - 7(0) = -12$$

$$4x = -12$$

$$x = \frac{-12}{4} = -3$$

Solution for б): $$x = -3, y = 0$$

System в)

The system of equations is:

• $$3x + 2y = 5$$
• $$-5x + 2y = 45$$

Step 1: Subtract the second equation from the first equation to eliminate y.

$$(3x + 2y) - (-5x + 2y) = 5 - 45$$

$$3x + 2y + 5x - 2y = -40$$

$$8x = -40$$

Step 2: Solve for x.

$$x = \frac{-40}{8} = -5$$

Step 3: Substitute the value of x into the first equation to solve for y.

$$3(-5) + 2y = 5$$

$$-15 + 2y = 5$$

$$2y = 5 + 15$$

$$2y = 20$$

$$y = \frac{20}{2} = 10$$

Solution for в): $$x = -5, y = 10$$

System г)

The system of equations is:

• $$9x - 4y = -13$$
• $$9x - 2y = -20$$

Step 1: Subtract the second equation from the first equation to eliminate x.

$$(9x - 4y) - (9x - 2y) = -13 - (-20)$$

$$9x - 4y - 9x + 2y = -13 + 20$$

$$-2y = 7$$

Step 2: Solve for y.

$$y = \frac{7}{-2} = -3.5$$

Step 3: Substitute the value of y into the first equation to solve for x.

$$9x - 4(-3.5) = -13$$

$$9x + 14 = -13$$

$$9x = -13 - 14$$

$$9x = -27$$

$$x = \frac{-27}{9} = -3$$

Solution for г): $$x = -3, y = -3.5$$

Final Answer:

• a) $$x = 5, y = -2$$
• б) $$x = -3, y = 0$$
• в) $$x = -5, y = 10$$
• г) $$x = -3, y = -3.5$$