Контрольные задания > Solve the following systems of linear equations: 1) { y = 2x, y = 6-x 2) { x+y = 0, 2x+2y = 2 3) { 2x-y = -1, x+y = -2
Вопрос:

Solve the following systems of linear equations: 1) { y = 2x, y = 6-x 2) { x+y = 0, 2x+2y = 2 3) { 2x-y = -1, x+y = -2

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Ответ:

Systems of Linear Equations

Here are the solutions to the given systems of linear equations:

System 1:

  • Equation 1: \( y = 2x \)
  • Equation 2: \( y = 6-x \)
Method: Since both equations are already solved for 'y', we can set them equal to each other to solve for 'x' and then substitute back to find 'y'. This is known as the substitution method.
  1. Step 1: Set the two equations equal to each other: \( 2x = 6-x \)
  2. Step 2: Solve for \( x \): \( 2x + x = 6 \) \( 3x = 6 \) \( x = 2 \)
  3. Step 3: Substitute the value of \( x \) into either original equation to find \( y \). Using the first equation: \( y = 2(2) \) \( y = 4 \)

Solution 1: \( x = 2, y = 4 \)

System 2:

  • Equation 1: \( x+y = 0 \)
  • Equation 2: \( 2x+2y = 2 \)
Method: We can use the elimination method or substitution method. Let's simplify the second equation first. If we divide the second equation by 2, we get \( x+y=1 \). Now we have two equations with the same left side but different right sides, indicating no solution.
  1. Step 1: From the first equation, we can express \( y \) as \( y = -x \).
  2. Step 2: Substitute this into the second equation: \( 2x + 2(-x) = 2 \)
  3. Step 3: Simplify: \( 2x - 2x = 2 \) \( 0 = 2 \)
  4. Step 4: This statement is false, which means there is no solution to this system of equations. The lines represented by these equations are parallel and distinct.

Solution 2: No solution

System 3:

  • Equation 1: \( 2x-y = -1 \)
  • Equation 2: \( x+y = -2 \)
Method: The elimination method is efficient here because the \( y \) terms have opposite coefficients. We can add the two equations together to eliminate \( y \).
  1. Step 1: Add the two equations: \( (2x-y) + (x+y) = -1 + (-2) \)
  2. Step 2: Simplify and solve for \( x \): \( 3x = -3 \) \( x = -1 \)
  3. Step 3: Substitute the value of \( x \) into either original equation to find \( y \). Using the second equation: \( -1 + y = -2 \)
  4. Step 4: Solve for \( y \): \( y = -2 + 1 \) \( y = -1 \)

Solution 3: \( x = -1, y = -1 \)

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