Solve systems of two linear equations in two variables by graphing, substitution, and elimination, and interpret one solution, no solution, or infinitely many solutions in context (A.EI.4).

What this topic is asking

A.EI.4 asks you to solve a system of two linear equations in two variables by graphing, substitution, and elimination, and to interpret the number of solutions. On the Virginia Algebra I SOL these are Equations and Inequalities items: solve and type an ordered pair, identify the number of solutions, or set up a system from a word problem. They appear as fill-in-the-blank, multiple choice, and coordinate-plane items.

What a system means

A system of equations is two (or more) equations considered together, and its solution is the set of values that make every equation true at the same time. For two lines, that is their point of intersection $(x, y)$. Solving the system is finding where the lines meet.

Substitution

Use substitution when one equation already has a variable isolated (like $y = 3x + 1$). Substitute that expression into the other equation, solve for the remaining variable, then back-substitute.

Elimination

Use elimination when the coefficients of one variable are equal or opposite, or can be made so by multiplying. Add or subtract the equations to cancel that variable.

For $3x + 2y = 16$ and $x - 2y = 0$: the $+2y$ and $-2y$ are opposites, so add the equations: $4x = 16$, $x = 4$. Then $4 - 2y = 0$ gives $y = 2$, so $(4, 2)$. If neither variable lines up, multiply one or both equations first (for example, double an equation so a $2y$ becomes $4y$ to match the other).

Graphing and the number of solutions

Graphing both lines and reading the crossing point works for whole-number intersections, and the Desmos calculator makes it quick. More importantly, the picture classifies the system:

• One solution. The lines have different slopes and cross at one point (a consistent, independent system).
• No solution. The lines have the same slope but different intercepts, so they are parallel and never meet (inconsistent).
• Infinitely many solutions. The two equations describe the same line (same slope and intercept), so every point on it works (consistent, dependent).

Why the algebra and the geometry agree

The three outcomes of a system are the same three outcomes you met when solving a single linear equation, for exactly the same reason. When you solve a system algebraically and the variables cancel, the leftover statement tells you which case you are in: a true statement (like $0 = 0$) means the equations are the same line, so infinitely many solutions; a false statement (like $0 = 5$) means the lines are parallel, so no solution. When the variables do not cancel, you get a unique value, so one solution. This mirrors the geometry precisely: identical lines overlap everywhere, parallel lines never touch, and lines with different slopes cross once. Picking a method (substitution, elimination, graphing) does not change the answer, only the route, because all three are asking the same question: which $(x, y)$ lies on both lines.

Modeling with a system

Word problems with two unknowns usually become a system. Name both unknowns, write one equation for each piece of information, then solve. "A school buys notebooks at \3$and folders at$\$2$, spending \60$on$25$items" gives$n + f = 25$and$3n + 2f = 60$. Substituting$n = 25 - f$leads to$f = 15$folders and$n = 10$notebooks; check$3(10) + 2(15) = 60$.

How the SOL examines this topic

• Fill-in-the-blank. Solve a system and type the ordered pair $(x, y)$.
• Multiple choice. Identify the number of solutions, or pick the solution.
• Coordinate-plane items. Graph two lines and select their intersection, or build a system from a context.

Try this

Q1. Solve $y = x + 1$ and $2x + y = 7$. [2 points]

• Cue. $2x + (x + 1) = 7 \Rightarrow 3x = 6 \Rightarrow x = 2$, $y = 3$; $(2, 3)$.

Q2. How many solutions does a system of two identical lines have? [1 point]

• Cue. The same line, so infinitely many.