Solutions by Substitution Flashcards

Substitution Method Overview

• The substitution method solves systems of equations by expressing one variable in terms of another and substituting it into a different equation.

Examples of Substitution in Equations

• For equations 2x + y = 7 and y = x + 1, substituting y yields 3x + 1 = 7.
• In equations 3x - y = 2 and y = x - 1, substitution produces 2x + 1 = 2.

Specific Solutions

• For 5x - 6y = 0 and y = x, the solution is (0, 0).
• The equations x - y = 6 and x = y + 2 have no solution indicating parallel lines.
• The system 10x - 10y = 1 and x = y - 3 also has no solution.

Further Substitution Results

• Substituting into 5x - 2y = 6 and x = 5 - y yields the solution (16/7, 19/7).
• The system x + y = 6 and 2x + y = 4 gives the solution (-2, 8).

Additional Solutions

• For equations 5x - 6 = y and 2x - 3y = 4, the solution is (14/13, -8/13).
• The equations 7x - 2 = 2y and 3x = 2y - 1 result in the solution (3/4, 13/8).
• Solving 8x = 2y + 5 and 3x = y + 7 yields (-9/2, -41/2).

Final Solutions from Substitution

• For 8y - 1 = x and 3x = 2y, the result is (1/11, 3/22).
• The system 7 + 2y = 8x and 3x - 2y = 0 has the solution (7/5, 21/10).

Identifying Equation Relationships

• Equations of the form 2x - y = c and x + 2y = d are identified as intersecting.
• For bx - ay = 2 and ax + by = 3, the equations are also intersecting.
• Given that b ≠ -1 in the equations x + y = ab and bx - y = a, they are classified as intersecting.