Solving systems of equations graphicallyIB MYP Maths Extended: Revision notes

Two linear equations such as $y=2x+1$ and $y=-x+7$ are called simultaneous equations. Each one is a straight line on a graph. A point on a line has $x$- and $y$-values that satisfy that line's equation. So the point where the lines intersect lies on both lines, and its coordinates satisfy both equations at once. That point is the solution.

1. Draw both lines on the same axes (use a table of values, or the gradient and $y$-intercept).
2. Find where the lines cross.
3. Read the coordinates $(x, y)$ of the intersection.
4. Check by substituting into both equations.\nFor $y=2x+1$ and $y=-x+7$ the lines meet at $(2,5)$. Check: $2(2)+1=5$ and $-2+7=5$.

To solve $f(x)=g(x)$ graphically, draw $y=f(x)$ and $y=g(x)$. The $x$-coordinates of the intersection points are the solutions. For $f(x)=x^2-2$ and $g(x)=x+4$, setting them equal gives $x^2-x-6=0$, so $(x-3)(x+2)=0$ and $x=3$ or $x=-2$. A line and a parabola can meet at two, one or no points. The $y$-coordinates come from substituting back: $(3,7)$ and $(-2,2)$.

Graphing software or a graphing calculator draws accurate graphs quickly. Type each function in, then use the intersect tool (or trace) to read the coordinates of the crossing points. Zooming in gives more accurate answers. Use technology for equations that do not have neat solutions, and say in your answer what you did, for example: the graphs meet at $x=3$ and $x=-2$. Round to the accuracy asked for.

Lines with the same gradient are parallel, so they never meet: there is no solution. If two equations describe the same line, every point on it is a solution. In real problems, the intersection tells you when two quantities are equal. For tank A, $V=50+5t$, and tank B, $V=20+8t$, the lines cross at $t=10$ minutes, $V=100$ litres: both tanks then hold the same volume. For $t<10$ tank A holds more water, and for $t>10$ tank B holds more. For hall hire, the crossing point shows the number of guests at which two companies cost the same, and which is cheaper before and after it.