Solve systems of linear equations algebraically (substitution and elimination) and graphically; solve a linear-quadratic system; create and solve systems from contexts; and graph the solution region of a system of linear inequalities.

What this topic is asking

The Regents Algebra I exam (the Reasoning with Equations and Inequalities, A-REI, and Creating Equations, A-CED, clusters) wants you to solve a system of two equations by substitution, elimination, or graphing; to solve a linear-quadratic system; to build a system from a context; and to graph the solution region of two linear inequalities. Systems word problems are among the most frequent constructed-response tasks on the exam.

Solving by substitution and elimination

Substitution works best when a variable is already isolated. Elimination works best when adding the equations cancels a variable.

Adding (1) and (2) eliminates $y$: $4x = 12$, so $x = 3$. Back-substitute into (2): $3 - 2y = -4$, so $-2y = -7$ and $y = 3.5$, giving $(3, 3.5)$. When no variable cancels directly, multiply an equation by a constant first so the coefficients of one variable are opposites.

A system can have one solution (lines cross), no solution (parallel lines, same slope, different intercept), or infinitely many (the same line). On the Regents this distinction is a common Part I item.

Linear-quadratic systems

When one equation is a line and the other a parabola, substitute the line into the quadratic and solve the resulting quadratic. For $y = x^2 - 4$ and $y = x + 2$, set $x^2 - 4 = x + 2$, so $x^2 - x - 6 = 0$, which factors to $(x - 3)(x + 2) = 0$, giving $x = 3$ and $x = -2$. The matching $y$-values are $5$ and $0$, so the system has two solutions, $(3, 5)$ and $(-2, 0)$. A linear-quadratic system can have two, one, or no real solutions, matching how often the line crosses the parabola.

Systems of inequalities

To graph a system of inequalities, graph each boundary line (dashed for $<$ or $>$, solid for $\leq$ or $\geq$), shade each half-plane, and the solution region is where the shadings overlap. A point is a solution only if it satisfies both inequalities, so the exam often asks you to test a labelled point such as $(2, 1)$ against the region.

A point worth stressing for the constructed-response credits is defining your variables before writing the system. Graders look for a clear statement such as "let $m$ be the number of muffins". A correct numerical answer with no defined variables and no shown system is capped below full credit, because the standard is about modeling, not just arithmetic.

Try this

Q1. Solve $y = x + 1$ and $y = 3x - 5$ by substitution. [2 credits]

• Cue. Set $x + 1 = 3x - 5$, so $6 = 2x$, $x = 3$, $y = 4$: the point $(3, 4)$.

Q2. How many solutions does $y = 2x + 1$ and $y = 2x - 4$ have? [1 credit]

• Cue. Same slope, different intercepts means parallel lines: no solution.