Question

1. \\( y = x - 4 \\)\\( 6x + 4y = 14 \\)8) \\( 4x - 2y = -6 \\)\\( y = 2x - 3 \\)9) \\( 7x - y = 8 \\)\\( y = -4x - 19 \\)10) \\( y = 2x + 3 \\)\\( -2x + y = 3 \\)11) \\( y = -x + 5 \\)\\( x - 4y = 10 \\)12) \\( x = y - 2 \\)\\( y - x = -5 \\)

Explanation:

Let's solve each system of equations one by one using the substitution method.

Problem 7:

System:
\( y = x - 4 \)
\( 6x + 4y = 14 \)

Step 1: Substitute \( y = x - 4 \) into the second equation.

Substitute \( y \) in \( 6x + 4y = 14 \) with \( x - 4 \):
\( 6x + 4(x - 4) = 14 \)

Step 2: Simplify and solve for \( x \).

Expand and combine like terms:
\( 6x + 4x - 16 = 14 \)
\( 10x - 16 = 14 \)
Add 16 to both sides:
\( 10x = 30 \)
Divide by 10:
\( x = 3 \)

Step 3: Substitute \( x = 3 \) back into \( y = x - 4 \) to find \( y \).

\( y = 3 - 4 = -1 \)

Problem 8:

System:
\( 4x - 2y = -6 \)
\( y = 2x - 3 \)

Step 1: Substitute \( y = 2x - 3 \) into the first equation.

Substitute \( y \) in \( 4x - 2y = -6 \) with \( 2x - 3 \):
\( 4x - 2(2x - 3) = -6 \)

Step 2: Simplify and solve for \( x \).

Expand and combine like terms:
\( 4x - 4x + 6 = -6 \)
\( 6 = -6 \)

This is a contradiction, so the system has no solution (parallel lines).

Problem 9:

System:
\( 7x - y = 8 \)
\( y = -4x - 19 \)

Step 1: Substitute \( y = -4x - 19 \) into the first equation.

Substitute \( y \) in \( 7x - y = 8 \) with \( -4x - 19 \):
\( 7x - (-4x - 19) = 8 \)

Step 2: Simplify and solve for \( x \).

Simplify the left side:
\( 7x + 4x + 19 = 8 \)
\( 11x + 19 = 8 \)
Subtract 19 from both sides:
\( 11x = -11 \)
Divide by 11:
\( x = -1 \)

Step 3: Substitute \( x = -1 \) back into \( y = -4x - 19 \) to find \( y \).

\( y = -4(-1) - 19 = 4 - 19 = -15 \)

Problem 10:

System:
\( y = 2x + 3 \)
\( -2x + y = 3 \)

Step 1: Substitute \( y = 2x + 3 \) into the second equation.

Substitute \( y \) in \( -2x + y = 3 \) with \( 2x + 3 \):
\( -2x + (2x + 3) = 3 \)

Step 2: Simplify.

Combine like terms:
\( 3 = 3 \)

This is an identity, so the system has infinitely many solutions (the same line).

Problem 11:

System:
\( y = -x + 5 \)
\( x - 4y = 10 \)

Step 1: Substitute \( y = -x + 5 \) into the second equation.

Substitute \( y \) in \( x - 4y = 10 \) with \( -x + 5 \):
\( x - 4(-x + 5) = 10 \)

Step 2: Simplify and solve for \( x \).

Expand and combine like terms:
\( x + 4x - 20 = 10 \)
\( 5x - 20 = 10 \)
Add 20 to both sides:
\( 5x = 30 \)
Divide by 5:
\( x = 6 \)

Step 3: Substitute \( x = 6 \) back into \( y = -x + 5 \) to find \( y \).

\( y = -6 + 5 = -1 \)

Problem 12:

System:
\( x = y - 2 \)
\( y - x = -5 \)

Step 1: Substitute \( x = y - 2 \) into the second equation.

Substitute \( x \) in \( y - x = -5 \) with \( y - 2 \):
\( y - (y - 2) = -5 \)

Step 2: Simplify.

Simplify the left side:
\( y - y + 2 = -5 \)
\( 2 = -5 \)

This is a contradiction, so the system has no solution (parallel lines).

Final Answers:

1. \( \boldsymbol{(x, y) = (3, -1)} \)
2. No solution
3. \( \boldsymbol{(x, y) = (-1, -15)} \)
4. Infinitely many solutions
5. \( \boldsymbol{(x, y) = (6, -1)} \)
6. No solution

Answer:

Let's solve each system of equations one by one using the substitution method.

Problem 7:

System:
\( y = x - 4 \)
\( 6x + 4y = 14 \)

Step 1: Substitute \( y = x - 4 \) into the second equation.

Substitute \( y \) in \( 6x + 4y = 14 \) with \( x - 4 \):
\( 6x + 4(x - 4) = 14 \)

Step 2: Simplify and solve for \( x \).

Expand and combine like terms:
\( 6x + 4x - 16 = 14 \)
\( 10x - 16 = 14 \)
Add 16 to both sides:
\( 10x = 30 \)
Divide by 10:
\( x = 3 \)

Step 3: Substitute \( x = 3 \) back into \( y = x - 4 \) to find \( y \).

\( y = 3 - 4 = -1 \)

Problem 8:

System:
\( 4x - 2y = -6 \)
\( y = 2x - 3 \)

Step 1: Substitute \( y = 2x - 3 \) into the first equation.

Substitute \( y \) in \( 4x - 2y = -6 \) with \( 2x - 3 \):
\( 4x - 2(2x - 3) = -6 \)

Step 2: Simplify and solve for \( x \).

Expand and combine like terms:
\( 4x - 4x + 6 = -6 \)
\( 6 = -6 \)

This is a contradiction, so the system has no solution (parallel lines).

Problem 9:

System:
\( 7x - y = 8 \)
\( y = -4x - 19 \)

Step 1: Substitute \( y = -4x - 19 \) into the first equation.

Substitute \( y \) in \( 7x - y = 8 \) with \( -4x - 19 \):
\( 7x - (-4x - 19) = 8 \)

Step 2: Simplify and solve for \( x \).

Simplify the left side:
\( 7x + 4x + 19 = 8 \)
\( 11x + 19 = 8 \)
Subtract 19 from both sides:
\( 11x = -11 \)
Divide by 11:
\( x = -1 \)

Step 3: Substitute \( x = -1 \) back into \( y = -4x - 19 \) to find \( y \).

\( y = -4(-1) - 19 = 4 - 19 = -15 \)

Problem 10:

System:
\( y = 2x + 3 \)
\( -2x + y = 3 \)

Step 1: Substitute \( y = 2x + 3 \) into the second equation.

Substitute \( y \) in \( -2x + y = 3 \) with \( 2x + 3 \):
\( -2x + (2x + 3) = 3 \)

Step 2: Simplify.

Combine like terms:
\( 3 = 3 \)

This is an identity, so the system has infinitely many solutions (the same line).

Problem 11:

System:
\( y = -x + 5 \)
\( x - 4y = 10 \)

Step 1: Substitute \( y = -x + 5 \) into the second equation.

Substitute \( y \) in \( x - 4y = 10 \) with \( -x + 5 \):
\( x - 4(-x + 5) = 10 \)

Step 2: Simplify and solve for \( x \).

Expand and combine like terms:
\( x + 4x - 20 = 10 \)
\( 5x - 20 = 10 \)
Add 20 to both sides:
\( 5x = 30 \)
Divide by 5:
\( x = 6 \)

Step 3: Substitute \( x = 6 \) back into \( y = -x + 5 \) to find \( y \).

\( y = -6 + 5 = -1 \)

Problem 12:

System:
\( x = y - 2 \)
\( y - x = -5 \)

Step 1: Substitute \( x = y - 2 \) into the second equation.

Substitute \( x \) in \( y - x = -5 \) with \( y - 2 \):
\( y - (y - 2) = -5 \)

Step 2: Simplify.

Simplify the left side:
\( y - y + 2 = -5 \)
\( 2 = -5 \)

This is a contradiction, so the system has no solution (parallel lines).

Final Answers:

1. \( \boldsymbol{(x, y) = (3, -1)} \)
2. No solution
3. \( \boldsymbol{(x, y) = (-1, -15)} \)
4. Infinitely many solutions
5. \( \boldsymbol{(x, y) = (6, -1)} \)
6. No solution