Skip to content

Systems of Equations

Definition

A system of equations is a set of two or more equations with the same variables. The goal is typically to find values for the variables that satisfy all equations in the system simultaneously. We often focus on systems of linear equations.

Example of a system of two linear equations with two variables: 

Equation 1: 2x + y = 5
Equation 2: x - y = 1
The solution to this system is the pair (x, y) that makes both equations true.

Prerequisites

To solve systems of equations, you need a solid understanding of:

  1. Linear Equations
  2. Basic Equation Solving
  3. Optionally: Linear Inequalities (for systems of inequalities)

Learning Objectives

After mastering systems of equations, you should be able to:

  1. Identify a system of equations.
  2. Understand what constitutes a solution to a system (a set of values satisfying all equations).
  3. Solve systems of linear equations using various methods:
    • Graphing (finding the intersection point)
    • Substitution
    • Elimination (or Addition/Subtraction method)
  4. Determine the number of solutions a system has (one unique solution, no solution, or infinitely many solutions) based on the equations or their graphs.
  5. Model real-world scenarios using systems of equations.
  6. Solve word problems by setting up and solving systems of equations.

Methods for Solving Systems of Linear Equations

Graphing

  • Graph each equation on the same coordinate plane.
  • The point where the lines intersect is the solution.
  • Limitations: Can be imprecise, difficult for non-integer solutions.

Substitution

  1. Solve one equation for one variable (e.g., solve Eq 1 for y).
  2. Substitute this expression into the other equation.
  3. Solve the resulting single-variable equation.
  4. Substitute the value found back into one of the original equations to find the other variable.

Elimination

  1. Multiply one or both equations by constants so that the coefficients of one variable are opposites (e.g., 3x and -3x).
  2. Add the modified equations together. One variable should cancel out (be eliminated).
  3. Solve the resulting single-variable equation.
  4. Substitute the value found back into one of the original equations to find the other variable.

Examples

Example 1: Solving by Substitution (Middle School/High School)

Problem: Solve the system: y = 2x + 1 3x + y = 11

Solution: 1. The first equation is already solved for y. 2. Substitute (2x + 1) for y in the second equation: 3x + (2x + 1) = 11 3. Solve for x: 5x + 1 = 11 5x = 10 x = 2 4. Substitute x = 2 back into y = 2x + 1: y = 2(2) + 1 y = 4 + 1 y = 5

Solution: (x, y) = (2, 5)

Example 2: Solving by Elimination (High School)

Problem: Solve the system: 2x + 3y = 7 4x - 3y = 5

Solution: 1. The y-coefficients are opposites (3 and -3). 2. Add the equations together: (2x + 3y) + (4x - 3y) = 7 + 5 6x = 12 3. Solve for x: x = 2 4. Substitute x = 2 back into the first equation 2x + 3y = 7: 2(2) + 3y = 7 4 + 3y = 7 3y = 3 y = 1

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

Example 3: Word Problem (High School / SAT)

Problem: A store sells large candles for $5 and small candles for $3. On Tuesday, they sold 20 candles in total and made $76. How many large candles were sold?

Solution: 1. Define variables: Let L be the number of large candles and S be the number of small candles. 2. Set up equations based on the information: * Total candles: L + S = 20 * Total revenue: 5L + 3S = 76 3. Solve the system. Using substitution: * From the first equation, S = 20 - L. * Substitute into the second equation: 5L + 3(20 - L) = 76 * 5L + 60 - 3L = 76 * 2L + 60 = 76 * 2L = 16 * L = 8

Answer: 8 large candles were sold.

Types of Solutions

  • One Unique Solution: Lines intersect at one point (different slopes).
  • No Solution: Lines are parallel (same slope, different y-intercepts). Leads to a contradiction like 0 = 5 when solving algebraically.
  • Infinitely Many Solutions: Lines are identical (same slope, same y-intercept). Leads to an identity like 0 = 0 when solving algebraically.

Common Misconceptions

  1. Substitution Errors: Substituting an expression back into the same equation it came from, instead of the other one.
  2. Elimination Errors: Forgetting to multiply the entire equation (both sides) by the constant, or making sign errors when adding/subtracting equations.
  3. Graphing Inaccuracies: Misinterpreting the intersection point, especially if it's not at integer coordinates.

Applications in SAT

Systems of equations are a major topic in the Heart of Algebra section. Questions may involve:

  1. Solving systems algebraically.
  2. Interpreting the number of solutions graphically or algebraically.
  3. Setting up systems from word problems.

Advanced Connections

Systems of linear equations are fundamental to:

  1. Matrices (representing and solving systems efficiently)
  2. Linear Algebra (vector spaces, linear transformations)
  3. Optimization (finding optimal solutions within constraints often defined by systems)

Practice Problems

  1. Basic (Graphing): Graph the system y = x + 2 and y = -x + 4 to find the solution.
  2. Intermediate (Substitution): Solve the system x = 3y - 1 and 2x + y = 5.
  3. Advanced (Elimination): Solve the system 3x - 2y = 8 and 2x + 5y = -1.
  4. SAT-Level: A Kiosk sells magazines ($4 each) and newspapers ($1 each). In the afternoon, they sold 50 items total for $110. How many magazines did they sell?