Solve systems by graphing and explain why the x-coordinates of intersections of y = f(x) and y = g(x) solve f(x) = g(x) (NC.M1.A-REI.11, A-REI.6).

What this topic is asking

NC.M1.A-REI.11 asks you to explain why the x-coordinates of the intersection points of $y = f(x)$ and $y = g(x)$ are the solutions of $f(x) = g(x)$, and to approximate solutions graphically or with technology. Combined with A-REI.6, this gives a graphical way to solve a system: the intersection point is the solution.

The intersection is the solution

A point on a line satisfies that line's equation; a point on both lines satisfies both.

Why intersections solve f(x) = g(x)

This is the explain-why heart of A-REI.11.

This idea also lets you solve any equation by graphing each side and finding where the graphs meet.

Reading a solution from a graph

To read a system's solution from a graph, find the crossing point and record its coordinates. If the crossing is not on a clean grid mark, you estimate, which is why A-REI.11 says "approximate." Technology (a graphing calculator or the on-screen NCTest calculator) gives a precise intersection.

The three cases

The slopes and intercepts decide the count:

• Different slopes: the lines cross once, one solution.
• Same slope, different y-intercept: parallel lines, no solution.
• Same slope and same y-intercept: the same line, infinitely many solutions.

How the NC Math 1 EOC examines this topic

• Technology-enhanced. Plot two lines and identify the intersection, or use the graphing tool to approximate a solution.
• Multiple choice. Choose the solution point, or the number of solutions from a description of slopes.
• Calculator-active. Graphical estimation fits the calculator-active section.

Graphing complements algebraic solving: the same answer appears as an intersection. The case analysis connects to equivalent systems and to comparing function families, where intersections of different function types are found graphically.

Why graphing and algebra must agree

Solving a system algebraically and solving it by graphing are two views of one fact: the solution is the pair that satisfies both equations. Algebra computes that pair exactly; the graph shows it as a crossing point. They must agree because they answer the identical question. This is reassuring and useful: a graph gives a quick visual estimate and a sanity check on an algebraic answer, while algebra gives the exact values a graph can only approximate. Holding both views, you can pick the right tool, exact algebra for precise answers, graphing for estimates and for equations too messy to solve by hand.

Try this

Q1. Where do $y = 2x$ and $y = x + 3$ intersect? [2 points]

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

Q2. Two lines have slopes $4$ and $4$ but different y-intercepts. How many solutions? [1 point]

• Cue. Parallel lines: no solution.