Variables

Question 1

x - 4y = 24
2x + 3y = 37

x = (24 + 4y) set single variable = rest

2(x) +3y = 37     then substitute for the (x)
2(24 +4y) + 3y = 37
48 +8y +3y = 37
48 +11y = 37
11y =37 - 48 = - 11
y = -11/11 

y = ( -1) then solve for other variable

Answer 1

Then solve for the other variable by substitution so
x - 4y = 24

y = (-1) then substitute this value into the original equation to solve for the (x)
x - 4(y) = 24 so that

x - 4(-1) = 24
x + 4 = 24
x = 24 - 4

x = ( 20)

(20, -1)

Question 2

2x - 4y = 40
3x + y = 25

3x + y = 25 set single variable = rest
y = (25 - 3x)

2x - 4(y) = 40   then substitute for the (y)
2x - 4(25 - 3x) = 40
2x - 100 + 12x = 40
14x - 100 = 40
14x = 140
x = 140/14 

x = (10) then solve for the other variable

Answer 2

Then solve for the other variable by substitution so
3(x) + y = 25

x = (10) then substitute this value into the original equation to solve for the (y)
3(x) + y = 25 so that

3(10) + y = 25
30 + y = 25
y = 25 - 30

y = (-5)

(10, -5)

Question 3

x + y = -3
x - y = 1

2x +0y = -2
2x = -2
x = -2/2

x = -1

Answer 3

Then solve for the other variable by substitution so
x + y = -3 so when x = (-1)

(-1) + y = -3
+1 +1

y = -3 + 1

y = -2

(-1, -2)

Question 4

3x - 5y = -16
2x + 5y = 31

5x + 0y = 15
5x = 15
x = 15/5

x = 3

Answer 4

Then solve for the other variable by substitution so
3x - 5y = -16 so when x = (3)

3(3) - 5y = -16
9 - 5y = - 16
- 9 -9

-5y = -25
y  = -25/-5

y = 5

(3, 5)

Question 5

3m - 2n = 13
m + 2n = 7

4m + 0n = 20
4m = 20
m = 20/4

m = 5

Answer 5

Then solve for the other variable by substitution so
3m - 2n = 13 so when m = (5)

3(5) - 2n = 13
15 - 2n = 13
-15 -15

-2n = -2
n = -2/-2

n = 1

(5, 1)

Question 6

-4x + 2y = 8
4x - 3y = -10

0x -y = -2
-y = -2
y = -2/-1

y = 2

Answer 6

Then solve for the other variable by substitution so
-4x +2y = 8 so when y = (2)

-4x +2(2) = 8
-4x +4 = 8
-4x = 8-4
-4x = 4
x = 4/-4

x = -1

(-1, 2)

Question 7

4x +5y = 7
8x +5y = 9 multiple by -1 to change signs

4x +5y = 7          then add equations
-8x -5y = -9     -1(8x + 5y) = -1(9)
-4x +0y = -2
-4x = -2
x = -2/-4

x = 0.5

Answer 7

Then solve for the other variable by substitution so
4x +5y = 7 so when x = (0.5)

4(0.5) +5y = 7
2 + 5y = 7
5y = 7-2
5y = 5
y = 5/5

y = 1

(0.5, 1)

Question 8

a - 2b = 5 multiply by -1 to change signs

3a - 2b = 9
-a + 2b = -5 then add equations
2a +0b = 4

2a = 4
a = 4/2

a = 2

Answer 8

Then solve for the other variable by substitution so
a - 2b = 5 so when a = (2)

(2) - 2b = 5
-2b = 5-2
-2b = 3
b = 3/-2

b = -1.5

(2, -1.5)

Question 9

5s + 2t = 6
9s + 2t = 22 multiple by -1 to change signs

5s + 2t = 6 then add equations

• 9s - 2t = -22
• 4s + 0t = -16

-4s = -16
s =  -16/-4

s = 4

Answer 9

Then solve for the other variable by substitution so
5s + 2t = 6 so when s = (4)

5(4) +2t = 6
20 + 2t = 6
2t = 6 - 20
2t = -14
t = -14/2

t= -7

(4, -7)

Question 10

x - 3y = 7
x + 2y = 2 multiple by -1 to change signs

x -3y = 7
-x -2y = -2 then add the equations
0x -5y = 5

-5y = 5
y = 5/-5

y = -1

Answer 10

Then solve for the other variable by substitution so
x - 3y = 7 when y = -1

x -3(-1) = 7
x +3 = 7
x = 7 - 3

x = 4

(4, -1)

Question 11

3x +4y = 6 pick a variable to change
5x +2y = -4 so you can eliminate 1 variable
multply by -2 to eliminate y
-2 (5x +2y = -4) > -10x -4y = 8 so

3x +4y = 6
-10x -4y = 8 then add the equations

-7x +0y = 14
-7x = 14
x = 14/-7

x = -2

Answer 11

Then solve for the other variable by substitution so
3x + 4y = 6 when x = (-2)

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

4y = 6 + 6
4y = 12

y = 12/4

y = 3

(-2, 3)

Question 12

-5x +3y = 6 pick a variable to change so you
x - y = 4 can eliminate 1 variable
multiply by 5 to eliminate x
5(x -y = 4) > 5x - 5y = 20 so

-5x +3y = 6
5x -5y = 20 then add the equations

0x -2y = 26
-2y = 26
y = 26/-2

y = -13

Answer 12

Then solve for the other variable by substitution so
x - y = 4 when y = (-13)

x -(-13) = 4
x +13 = 4
x = 4 -13

x = -9

(-9, -13)

Question 13

x + y = 3 pick a variable to change so you
2x - 3y = 16 can eliminate 1 variable
multiply by -2 to eliminate x
-2( x + y = 3) > -2x - 2y = -6

-2x - 2y = -6
2x - 3y = 16 then add the equations

0x - 5y = 10
5y = 10
y = 10 / 5

y = -2

Answer 13

Then solve for the other variable by substitution so
x + y = 3 when y = (-2)

x +(-2) = 3
x - 2 = 3
x = 3 + 2

x = 5

(5, -2)

Question 14

2x - y = 6 pick a variable to change so you
3x + 4y = -2 can eliminate 1 variable
Multiply by 4 to eliminate y

4(2x - y = 6) > 8x -4y = 24

8x - 4y = 24
3x + 4y = -2 then add the equations

11x +0y = 22
11x = 22
x = 22/11

x = 2

Answer 14

Then solve for the other variable by substitution so
3x +4y = -2 when x = (2)

3(2) +4y = -2
6 + 4y = -2 
4y = -2 - 6
4y = -8
y = -8/4

y = -4

(2, -4)

Question 15

2x - y = 6 pick a variable to change so you
3x + 4y = -2 can eliminate 1 variable
multipy by 4 to eliminate y

4(2x - y = 6) > 8x -4y = 24

8x - 4y = 24
3x + 4y = -2 then add the equations

11x +0y = 22
11x = 22
x = 22/11

x = 2

Answer 15

Then solve for the other variable by substitution so
3x +4y = -2 when x = (2)

3(2) +4y = -2
6 + 4y = -2 
4y = -2 - 6
4y = -8
y = -8/4

y = -2

(2, -2)

Question 16

x + 5y = 4 Pick a variable to change so you
3x - 7y = -10 can eliminate 1 variable
Multiply by -3 to eliminate x
-3(x + 5y = 4) > -3x - 15y = -4

-3x -15y = -12
3x - 7y = -10 then add the equations

0x - 22y = -22
-22y = -22
y = -22/-22

y = 1

Answer 16

Then solve for the other variable by substitution so
x + 5y = 4 when y = (1)

x + 5(1) = 4
x + 5 = 4

x = 4 - 5

x = -1

(-1 , 1)

Question 17

2x + y = 5 Pick a variable to change so you
3x - 2y = 4 can eliminate 1 variable
Multiply by 2 to eliminate y
2(2x + y = 5) > 4x + 2y = 10

4x + 2y = 10
3x - 2y = 4 then add the equations

7x + 0y = 14
7x = 14
x = 14/7

x = 2

Answer 17

Then solve for the other variable by substitution so
2x + y = 5 when x = (2)

2(2) + y = 5
4 + y = 5
y = 5 - 4

y = 1

(2, 1)

Question 18

Steps  to Substitution
1. Get Variable alone
      ~ Undo all addition/subtraction
      ~ Undo all multiplying/dividing
2. Take that new value for the variable and 
         plug into second equation.
3. Solve for the variable
      ~ Combine all like terms
      ~ Undo all addition/subtraction
      ~ Undo all multiplying/dividing
4. Take the new value for that variable and
         plug into either original equation.
5. Solve

Answer 18

x +4y = 8 x = (8 -4y)
2x -5y = 29

2(8 -4y) -5y = 29
16 - 8y - 5y = 29
16 -13y = 29
-13y = 29 -16 = 13
y = 13/-13 = -1

x +4(-1) = 8
x - 4 = 8
x = 8 + 4 = 12

(12, -1)

Question 19

Steps to Elimination
1. Line up the Variables
2. See if anything cancels
3. Change sign and or value to cancel
      ~ multiply/divide by -1 
      ~ multiply/divide to get  
4. Add the equations to cancel
5. Solve
6. Take the new value for that variable and
        plug into either original equation
        to find the other variable

Answer 19

x + y = 41
x - y = 5

2x + 0 = 46
x = 46/2 = 23

x + y = 41
23 + y = 41
y = 41 - 23 = 18

(23, 18)

Question 20

Steps to Graphing
1. Change equation to y = mx +b format
2. m = slope of the line / \
3. b = y-intercept = value of y when x 
         is zero [always located on the
         Y-axis]
4. Identify Solutions
      ~ one solution
      ~ many solutions
      ~ No solution

Answer 20

-4x + y = 3

y = mx + b
y = 4x + 3

m = 4 slope 4 up 1 over in positive direction
b = 3 , so when x = 0, y = 3    (0, 3)