Solve systems of two linear equations in two variables by graphing, substitution, and elimination, and interpret the solution, including consistent, inconsistent, and dependent systems (MA.912.AR.9.1, MA.912.AR.9.4).

What this topic is asking

MA.912.AR.9 asks you to solve a system of two linear equations by graphing, substitution, and elimination, and to interpret the result. The solution is the point where the lines meet. The B.E.S.T. Algebra 1 EOC tests all three methods, plus the modeling of real situations as two equations, with equation-editor and context items.

Three methods

Graphing plots both lines; the intersection is the solution. It is visual but imprecise (and the EOC calculator does not graph), so it is best for whole-number answers.

Substitution solves one equation for a variable and plugs it into the other:

Elimination adds or subtracts the equations (after scaling) so one variable cancels.

One, none, or infinitely many

The geometry of two lines gives three cases:

• One solution (consistent, independent): lines cross once, different slopes.
• No solution (inconsistent): parallel lines, same slope, different intercepts; the variables cancel to a false statement.
• Infinitely many (dependent): the same line; the variables cancel to a true statement.

How the B.E.S.T. EOC examines this topic

• Equation editor. Solve a system and enter the ordered pair.
• Context items. Set up and solve a system from a word problem (tickets, mixtures, two rates).
• Multiple choice and matching. Identify the number of solutions or match a system to its graph.

A clarifying idea: solving a system asks where two relationships agree at the same time. The single point answer is the one $(x, y)$ that makes both equations true together, which is why you must check it in both equations, not just one.

Why elimination works

Elimination relies on a property you already trust for single equations: adding equal quantities to both sides keeps an equation true. If $2x - y = 4$ is true, then adding $2x - y$ to one side of another equation and $4$ to the other side adds the same amount to both, preserving truth. By scaling one or both equations so a variable has matching coefficients, that variable cancels when you add or subtract, collapsing two equations into one with a single variable. The reason the result is still the system's solution is that any $(x, y)$ satisfying both originals also satisfies their sum or difference, so no solutions are gained or lost. Choosing which variable to eliminate, pick the one whose coefficients are already equal or easy to match, is what makes the method fast.

Choosing a method

Use substitution when one equation already has a variable isolated (like $y = \dots$). Use elimination when the coefficients of one variable are equal or opposite, or easy to scale to match. Use graphing only for quick whole-number intersections. All three give the same answer; speed comes from matching the method to the form.

Try this

Q1. Solve $y = x - 2$ and $2x + y = 10$. [2 points]

• Cue. $2x + (x - 2) = 10 \Rightarrow 3x = 12 \Rightarrow x = 4$, $y = 2$; $(4, 2)$.

Q2. Two lines have the same slope but different $y$-intercepts. How many solutions? [1 point]

• Cue. None (parallel, inconsistent).