Systems of Equations Review

Solving Systems of Equations Algebraically

Substitution Method

• Solve one equation for one variable.
• Substitute that expression into the other equation.
• Solve for the remaining variable.
• Substitute the value back to find the other variable.

Elimination Method

• Multiply one or both equations by a constant so that one variable has the same coefficient but opposite signs.
• Add the equations to eliminate that variable.
• Solve for the remaining variable.
• Substitute the value back to find the other variable.

Types of Solutions

• One Solution: The lines intersect at one point.
• No Solution: The lines are parallel and do not intersect.
• Infinitely Many Solutions: The lines are coincident (the same line).

Word Problems

• Define variables.
• Write a system of equations based on the problem.
• Solve the system using substitution or elimination.
• Interpret the solution in the context of the problem.

Examples

• Substitution Example:
  • $y = 5x$
  • $3x - 2y = 14$
  • $3x - 2(5x) = 14$
  • $3x - 10x = 14$
  • $-7x = 14$
  • $x = -2$
  • $y = 5(-2) = -10$
  • Solution: $(-2, -10)$
• Elimination Example:
  • $x + 2y = -7$
  • $x - 5y = 7$
  • Subtract the second equation from the first:
  • $(x + 2y) - (x - 5y) = -7 - 7$
  • $7y = -14$
  • $y = -2$
  • Substitute back: $x + 2(-2) = -7$
  • $x - 4 = -7$
  • $x = -3$
  • Solution: $(-3, -2)$

Determining the Number of Solutions Without Solving

• Rewrite equations in slope-intercept form ($y = mx + b$).
• Compare slopes and y-intercepts:
  • Same slope, different y-intercepts: No solution (parallel lines).
  • Same slope, same y-intercept: Infinitely many solutions (coincident lines).
  • Different slopes: One solution (intersecting lines).

Common Mistakes

• Distributing negatives properly when using elimination.
• Substituting the value back into the correct equation to solve for the other variable.
• Interpreting word problems correctly to set up the system of equations.