Systems of Linear Equations

A system is two (or more) linear equations about the same situation. Each equation is a line. A solution is an ordered pair $(x, y)$ that makes both equations true at once.

How to do it:

1. Treat each equation as its own line.

2. A solution must sit on every line.

3. One true equation is not enough.

Worked example: $(2, 7)$ solves $y = 2x + 3$ because $7 = 4 + 3\text{.}$ It also needs to satisfy the second equation before it is a system solution.

Check: a point on the first line only is not a system solution. That is why "it makes the first true" is not the definition.

Substitution isolates a variable in one equation, then replaces that variable in the other. You get one equation in one variable, solve it, then back-substitute.

How to do it:

1. Solve one equation for $x$ or $y$ (or use a form that already says $y = \ldots\text{).}$

2. Replace that variable in the other equation.

3. Solve, then plug back to find the partner value. Check in both originals.

Worked example: $y = 2x + 3$ and $4x + y = 19\text{.}$

Then $y = 2\left(\dfrac{8}{3}\right) + 3 = \dfrac{25}{3}\text{.}$

Check: if one equation already says $x = 4\text{,}$ substitute $4$ immediately. That is faster than elimination. Replay a one-variable check in the Equation Solver.

Elimination adds or subtracts whole equations so one variable cancels. You may multiply an equation by a constant first so coefficients match or become opposites.

How to do it:

1. Line up like terms.

2. Multiply one or both equations so one variable's coefficients match or oppose.

3. Add or subtract the equations, solve what remains, then back-substitute.

Worked example: $x + y = 5$ and $x - y = 1\text{.}$ Add:

Then $3 + y = 5\text{,}$ so $y = 2\text{.}$

Check: $3 - 2 = 1\text{.}$ Both originals work. Multiplying only one side of an equation would break the balance; scale the whole equation.

Graph both lines on the same axes. The intersection is the solution, because that point lies on both lines.

How to do it:

1. Write each equation as $y = mx + b$ if you need to plot.

2. Graph both lines.

3. Read the intersection. If the crossing is messy, use substitution or elimination for an exact pair.

Worked example: $y = 2x + 1$ and $y = -x + 4$ meet when $2x + 1 = -x + 4\text{,}$ so $3x = 3\text{,}$ $x = 1\text{,}$ $y = 3\text{.}$ The graph should cross at $(1, 3)\text{.}$

Check: the $y$-intercept of either line is not the solution unless both lines meet there. Plot both in the Function lab or the Graphing Calculator.

• One solution: the lines cross once.

• No solution: distinct parallel lines; the system is inconsistent.

• Infinitely many: both equations describe the same line; the system is dependent. Elimination can leave $0 = 0\text{.}$

How to do it:

1. Compare slopes and intercepts, or simplify both equations.

2. Same slope, different intercept: no solution.

3. One equation is a multiple of the other: infinitely many.

Worked example: $y = 2x + 1$ and $y = 2x + 5$ never meet. $2x + y = 6$ and $4x + 2y = 12$ are the same line, so every point on that line works.

Check: $0 = 0$ after elimination means infinitely many, not a mistake. $0 = 5$ means no solution.

Each sentence becomes an equation. After you solve, check that the pair fits both stories, with units.

How to do it:

1. Define both variables in a sentence.

2. Write one equation for a count or total, and one for a money or rate total.

3. Solve, then say the answer in the story.

Worked example: Adult tickets are $\$12\text{,}$ student tickets are $\$7\text{,}$ $200$ tickets sold, revenue $\$1775\text{.}$ Let $a$ be adults and $s$ be students.

Check: swapping the $200$ and $1775$ would mix counts with dollars. The streaming-bundle story uses the same move: $12 + 18m = 24m$ is two cost rules set equal.