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# Systems of Linear Equations Substitution Solutions

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### 1. Solution: Infinite Solutions

y = 1 + 5x
12y – 12 = 60x

1. Since y is isolated in the top equation, replace y in the bottom equation with 1 + 5x.

12(1 + 5x) - 12 = 60x

2. Simplify and solve.

12 + 60x - 12 = 60x
0 + 60x = 60x
60x = 60x

Because 60x = 60x is the result, there are infinite solutions in this system of equations. In other words, these 2 equations represent the same line:

Equation 1: y = 1 + 5x

Solve for y in Equation 2:

12y-12 = 60x
12y - 12 + 12 = 60x + 12
12y + 0 = 60x + 12
12y = 60x + 12
12y/12 = 60x/12 + 12/12
y = 5x + 1

Thus, these two lines are the same.

### 2. Solution: (4, -2)

x = 4
3x + y = 10

1. Since you already know that x = 4, plug in 4 for x in the bottom equation.

3x + y = 10
3(4) + y = 10

2. Simplify and solve.

12 + y = 10
12 + y + - 12 = 10 + -12
y = -2

3. Verify that (4, -2) is the correct answer.

3x + y = 10
3(4) + -2 = 10
12 + -2 = 10
10 = 10

### 3. Solution: (-5, 8)

x = -7 + ¼y
y = -2x – 2

1. Wherever you see an x in the bottom equation, plug in -7 + ¼y.

y = -2x – 2
y
= -2(-7 + ¼y) – 2

2. Simplify and solve.

y = 14 + -½y - 2
y = 12 + -½y
y + ½y = 12 + -½y + ½y

1y + ½y = 12 + 0
y= 12
y/1½= 12/1½

y = 8

3. Plug in y = 8 and solve for x.

x = -7 + ¼y
x = -7 + ¼(8)
x = -7 + 2
x = -5

4. Verify that (-5, 8) is the solution.

y = -2x – 2
8 = -2(-5) - 2
8 = 10 -2
8 = 8

### 4. Solution: (-5.5, -11.5)

y = 3x + 5
xy = 6

1. Since y is isolated in the top equation, substitute 3x + 5 for y in the bottom equation.

xy = 6
x
(3x + 5) = 6

2. Simplify and solve.

x + -3x + -5 = 6
-2x + -5 = 6
-2x + -5 + 5 = 6 + 5

-2x = 11
-2x/-2 = 11/-2

x= -5.5

3. Plug in x = -5.5 and solve for y.

y = 3x + 5
y = 3(-5.5) + 5
y = -16.5 + 5
y = -11.5

4. Verify that (-5.5, -11.5) is the solution.

xy = 6
-5.5 - -11.5 = 6
-5.5 + 11.5 = 6
6 = 6

### 5. Solution: (3/7, 29/7)

y = 5x + 2
y + 2x = 5

1. Since y is isolated in the top equation, substitute 5x + 2 for y in the bottom equation.

y + 2x = 5
5x + 2 + 2x = 5

2. Simplify and solve.

7x + 2 = 5
7x + 2 + -2 = 5 + -2

7x = 3
7x/7 = 3/7

x = 3/7

3. Plug in x = 3/7 and solve for y.

y = 5x + 2
y= 5(3/7) + 2

y= 15/7 + 2
y = 29/7

4. Verify that (3/7, 29/7) is correct.

y + 2x = 5
29/7 + 2(3/7) = 5
29/7 + 6/7 = 5

35/7 = 5

5 = 5