Solving Systems of Linear Equations

TL;DR

You'll learn how to find values for variables that satisfy multiple linear equations simultaneously. We'll use methods like substitution and elimination to systematically reduce equations and isolate variables. The goal is to find a unique solution, infinitely many solutions, or determine if no solution exists.

1. The Mental Model

Think of a system of linear equations as a set of rules. Each equation is a rule that links your variables. When you "solve" the system, you're looking for values for those variables that obey all the rules at the same time.

2. The Core Material

Solving systems of linear equations means finding values for variables that make all equations in the system true. Geometrically, each linear equation represents a line (in 2D), a plane (in 3D), or a hyperplane (in higher dimensions). The solution is where these lines, planes, or hyperplanes intersect.

There are a few main ways to solve these systems:

Substitution Method

This method is great when one of your equations can easily be rearranged to express one variable in terms of others.

1. Solve for one variable: Pick one equation and rearrange it to isolate one variable (e.g., $x = \text{something with } y$).
2. Substitute: Take that expression and substitute it into the other equation(s). This reduces the number of variables in the equation(s) you just substituted into.
3. Solve the new equation: Now you have an equation with fewer variables. Solve it.
4. Back-substitute: Take the value(s) you found and substitute them back into one of the original equations (or your rearranged expression) to find the remaining variable(s).

Elimination Method (or Addition/Subtraction)

This method aims to eliminate one variable by adding or subtracting multiples of the equations.

1. Align variables: Write the equations so that like variables are stacked vertically.
2. Multiply to match coefficients: Choose a variable to eliminate. Multiply one or both equations by constants so that the coefficients of that chosen variable are either the same or additive inverses (e.g., $3x$ and $-3x$).
3. Add or subtract: Add or subtract the equations to eliminate the chosen variable.
4. Solve the new equation: Solve the resulting equation, which now has fewer variables.
5. Back-substitute: Substitute the value(s) you found back into one of the original equations to find the remaining variable(s).

What the solutions look like

There are three possible outcomes when solving a system of linear equations:

graph TD
    A["System of Linear Equations"] --> B{"Solve using Substitution or Elimination"};
    B --> C["Unique Solution (e.g., x=2, y=3)"];
    B --> D["Infinitely Many Solutions (e.g., lines are identical)"];
    B --> E["No Solution (e.g., lines are parallel)"];

• Unique Solution: You find a single value for each variable. Geometrically, the lines/planes intersect at exactly one point.
• Infinitely Many Solutions: The equations are essentially the same (one is a multiple of the other). Geometrically, the lines/planes perfectly overlap. When you try to solve, you'll end up with a true statement like $0 = 0$.
• No Solution: The equations are contradictory. Geometrically, the lines/planes are parallel and never intersect. When you try to solve, you'll end up with a false statement like $0 = 5$.

3. Worked Example

Let's solve the following system using both substitution and elimination:

Equation 1: $2x + y = 7$
Equation 2: $x - 3y = 0$

Using Substitution:

1. Solve for one variable: From Equation 2, it's easy to get $x$:
   $x = 3y$

2. Substitute: Substitute $x = 3y$ into Equation 1:
   $2(3y) + y = 7$
   $6y + y = 7$

3. Solve the new equation:
   $7y = 7$
   $y = 1$

4. Back-substitute: Substitute $y = 1$ back into $x = 3y$:
   $x = 3(1)$
   $x = 3$

The solution is $(x, y) = (3, 1)$.

Using Elimination:

1. Align variables:
   $2x + y = 7$
   $x - 3y = 0$

2. Multiply to match coefficients: Let's eliminate $y$. Multiply Equation 1 by 3:
   $3(2x + y) = 3(7) \Rightarrow 6x + 3y = 21$
   Equation 2 remains: $x - 3y = 0$

3. Add or subtract: Add the two new equations:
   $(6x + 3y) + (x - 3y) = 21 + 0$
   $7x = 21$

4. Solve the new equation:
   $x = 3$

5. Back-substitute: Substitute $x = 3$ into original Equation 2:
   $3 - 3y = 0$
   $-3y = -3$
   $y = 1$

Again, the solution is $(x, y) = (3, 1)$. Both methods give the same result!

4. Key Takeaways

• Solving linear systems means finding variable values that satisfy all equations simultaneously.
• The substitution method works by isolating a variable and plugging its expression into other equations.
• The elimination method works by manipulating equations to cancel out a variable when added or subtracted.
• A system can have a unique solution, infinitely many solutions, or no solution at all.
• If you get a true statement like $0=0$, there are infinitely many solutions.
• If you get a false statement like $0=5$, there is no solution.
• Always check your solution by plugging the values back into the original equations.

5. Now Try It

Solve the following system of equations using either substitution or elimination (your choice!):

$3x - 2y = 11$
$x + 5y = -3$

What to do: Find the values of $x$ and $y$. What success looks like: You should find a unique pair of $(x, y)$ values that make both equations true.