Solve the following systems of equations using the substitution method.

Task: Solve systems of equations using the substitution method.

System 1a:

• Equation 1: \( 3x - y = 3 \)
• Equation 2: \( 3x - 2y = 0 \)

Step-by-step solution:

1. Step 1: From Equation 1, isolate y: \( y = 3x - 3 \).
2. Step 2: Substitute this expression for y into Equation 2: \( 3x - 2(3x - 3) = 0 \).
3. Step 3: Simplify and solve for x: \( 3x - 6x + 6 = 0 \) \( -3x = -6 \) \( x = 2 \).
4. Step 4: Substitute the value of x back into the expression for y: \( y = 3(2) - 3 \) \( y = 6 - 3 \) \( y = 3 \).

Answer for 1a: x = 2, y = 3

System 2a:

• Equation 1: \( x + 5y = 7 \)
• Equation 2: \( 3x + 2y = -5 \)

Step-by-step solution:

1. Step 1: From Equation 1, isolate x: \( x = 7 - 5y \).
2. Step 2: Substitute this expression for x into Equation 2: \( 3(7 - 5y) + 2y = -5 \).
3. Step 3: Simplify and solve for y: \( 21 - 15y + 2y = -5 \) \( -13y = -26 \) \( y = 2 \).
4. Step 4: Substitute the value of y back into the expression for x: \( x = 7 - 5(2) \) \( x = 7 - 10 \) \( x = -3 \).

Answer for 2a: x = -3, y = 2

System 3a:

• Equation 1: \( x + y = 7 \)
• Equation 2: \( 5x - 7y = 11 \)

Step-by-step solution:

1. Step 1: From Equation 1, isolate x: \( x = 7 - y \).
2. Step 2: Substitute this expression for x into Equation 2: \( 5(7 - y) - 7y = 11 \).
3. Step 3: Simplify and solve for y: \( 35 - 5y - 7y = 11 \) \( -12y = -24 \) \( y = 2 \).
4. Step 4: Substitute the value of y back into the expression for x: \( x = 7 - 2 \) \( x = 5 \).

Answer for 3a: x = 5, y = 2

System 4a:

• Equation 1: \( 6x + y = 5 \)
• Equation 2: \( 2x - 3y = -5 \)

Step-by-step solution:

1. Step 1: From Equation 1, isolate y: \( y = 5 - 6x \).
2. Step 2: Substitute this expression for y into Equation 2: \( 2x - 3(5 - 6x) = -5 \).
3. Step 3: Simplify and solve for x: \( 2x - 15 + 18x = -5 \) \( 20x = 10 \) \( x = 1/2 \).
4. Step 4: Substitute the value of x back into the expression for y: \( y = 5 - 6(1/2) \) \( y = 5 - 3 \) \( y = 2 \).

Answer for 4a: x = 1/2, y = 2

System 5a:

• Equation 1: \( 4x - 3y = -1 \)
• Equation 2: \( x - 5y = 4 \)

Step-by-step solution:

1. Step 1: From Equation 2, isolate x: \( x = 4 + 5y \).
2. Step 2: Substitute this expression for x into Equation 1: \( 4(4 + 5y) - 3y = -1 \).
3. Step 3: Simplify and solve for y: \( 16 + 20y - 3y = -1 \) \( 17y = -17 \) \( y = -1 \).
4. Step 4: Substitute the value of y back into the expression for x: \( x = 4 + 5(-1) \) \( x = 4 - 5 \) \( x = -1 \).

Answer for 5a: x = -1, y = -1