Quick Math help Solution sets?

Question

What is the solution of
1. 4y=x+4
x=y+2

2. 2x-y=4
4x-2y=4

Solve using the substitute method
3. 7x+8y=-25
x=-1-2y

4. 9x-3y=6
3x-2=y

5. 2x+4y=-5
2x-y=7

Solve Using the elimination method

6. 8x+5y=-9
x-10y=-33

Answer 1

1).

Substitution method :

The system of equations are 4y = x + 4 and x = y + 2.

Substitute x = y + 2 in equation 1 : 4y = x + 4.

4y = (y + 2) + 4

4y = y + 2 + 4

4y - y = 6

3y = 6

⇒ y = 6/3 = 2.

Substitute the value y = 2 in either of equation for x.

equation 2 : x = y + 2.

x = 2 + 2

⇒ x = 4.

The solution of the given system is x = 4 and y = 2.

2).

The system of equations are 2x - y = 4 ----------> (1)

                                            4x - 2y = 4 ----------> (2)

Divide equation 2 : 4x - 2y = 4 by 2.

2x - y = 2.

Observe the two lines i.e, 2x - y = 4 and 2x - y = 2, they have same slopes and different y - intercepts.

So, the given lines are parallel lines.

Since, the given lines are parallel lines, the system has no solution.

Substitution method :

3).

The system of equations are 7x + 8y = - 25 and x = - 1 - 2y.

Substitute x = - 1 - 2y in equation 1 : 7x + 8y = - 25.

7(- 1 - 2y) + 8y = - 25

- 7 - 14y + 8y = - 25

- 7 - 6y = - 25

6y = 25 - 7 = 18

y = 18/6

⇒ y = 3.

Substitute the value y = 3 in either of equation for x.

equation 2 : x = - 1 - 2y.

x = - 1 - 2(3)

x = - 1 - 6

⇒ x = - 7.

The solution of the given system is x = - 7 and y = 3.

4).

The system of equations are 9x - 3y = 6 ----------> (1)

                                             3x - 2 = y    ----------> (2)

Divide equation 1 : 9x - 3y = 6 by 3.

3x - y = 2.

Add y to each side.

3x = 2 + y

Subtract 2 from each side.

3x - 2 = y.

Observe the two lines i.e, 3x - 2 = y and 3x - y = 2, they have same slopes and same y - intercepts.

So, the given lines are equal lines.

Since, the given lines are equal lines, the system has infinitely many solutions.

Answer 2

Contd........

5).

The system of equations are 2x + 4y = - 5 and 2x - y = 7.

Simplify equation 2 : 2x - y = 7.

y = 2x - 7.

Substitute y = 2x - 7 in equation 1 : 2x + 4y = - 5.

2x + 4(2x - 7) = - 5

2x + 8x - 28 = - 5

10x = - 5 + 28 = 23

⇒ x = 23/10.

Substitute the value x = 23/10 in either of equation for y.

equation 2 : 2x - y = 7.

2(23/10) - y = 7

y = (23/5) - 7

y = (23 - 35)/5

⇒ y = - 12/5.

The solution of the given system is x = 23/10 and y = - 12/5.

6).

Elimination method :

The system of equations are 8x + 5y = - 9 -----------> ( 1 )

                                             x - 10y = - 33 -----------> ( 2 )

To eliminate the y - variable, multiply equation (1) by 2, then write the equations in column form, then add the equations.

16x + 10y = - 18

x - 10y = - 33

( + )_______________

17x = - 51

⇒ x = - 51/17 = - 3.

Substitute the value x = - 3 in either of equation.

Equation (2) : x - 10y = - 33.

- 3 - 10y = - 33

10y = 33 - 3 = 30

⇒ y = 30/10 = 3.

The solution of the given system is x = - 3 and y = 3.