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

When solving systems of equations using the substitution method,  first find the equation that has a variable with a coefficient of 1, if possible.  Then solve for that variable.

 

1. You have an equation that is X = y + 2.  Substitute y + 2 in place of the x in the other equation.

4y = y + 2 +4----- 4y - y = 6-----3y = 6------y = 2

 

Now that you know the value of y, substitute that value in place of the y in the equation x = y + 2.

X = 2 + 2;  X = 4. 

 

The solution is (4, 2)  or x = 4 and y = 2. 

 

You can check this solution by substituting these values in for x and y in both equations and ensuring that both sides are equal to each other.

 

4(2) = 4 + 4------ 8=8.  And  4 = 2+2------4 = 4.

 

2.  We will solve the equation 2x - y = 4 for y, because it is the variable with a coefficient of 1.  First, move the y to the other side of the equal sign, so it will be postive.

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

 

Now, substitute 2x - 4 in place of the y in the other equation and solve for x.

 

4x - 2 (2x - 4) = 4----- 4x -4x + 8 = 4.  As you can see, you are going to have a zero value for x, therefore you will end up with 0 = -4, which is not a true statement.  This means there are no solutions for this system of equations.  

 

The solution to this problem is: no solution.

 

3.  X = -1 - 2y----- 7(-1 -2y) + 8y = - 25------ -7 -14y + 8y = -25----- -6y = -25 + 7 ---- -6y = - 18---  y = 3

X = -1 -2 (3)----  -1 -6-----  -7.  Solution (-7, 3)  or x = -7 and y = 3

 

4.  Y = 3x - 2----- 9x -3 (3x -2) = 6------ 9x - 9x + 6 = 6---- 0 = 6 -6 --- 0=0

Becuase the x and the number value are both zero, you end up with 0 = 0, which is a true statement.  Therefore,  the solution to this problem is all real numbers.  You can test it out put any number in place of the x and y in either equation and you will see you will end up with a true statement.

 

Solution is:  all real numbers

 

5.  Y = 2x - 7----- 2x + 4 (2x - 7) = -5------ 2x + 8x - 28 = -5-----10x = -5 + 28--- 10 x = 23---x = 23/10.

 

Y = 2(23/10) -7. Y = 23/5 -7. Y = (23 - 35)/5.  Y = -12/5

 

Solution is (23/10, -12/5)  or x = 23/10 and y = - 12/5.

 

6.when using the elimination method you multiply one equation by a number that will give you a value the will eliminate either the x or y value in the othe equation.  In this case we have one equation that has a 5y and the other equation has a - 10y, so we will multiply the entire equation 8x + 5y = -9 by 2, then we will add the two equations and solve for x.

 

2 ( 8x + 5y) = 2 (-9)----- 16x + 10y = -18. 

 

(16x + 10y = - 18) + (x - 10y = -33)---add all like terms 16x + x = 17 x;  10y - 10y = 0; -18 -33, resulting 17x = -51.  Now ddivide both sides by 17 to solve for x.  X = -3.  

 

Now substitute -3 for x in the equation (x -10y = -33) and solve for y.

 

-3 -10y = -33------  -3 + 33 = 10y ---- 30 = 10y----- y = 3.

 

The solution is (-3,3)  or x = -3 and y = 3.