Solving equations with graphs

The solution of two simultaneous equations is where their graphs cross: that point fits both equations.

1. Draw both lines on the same axes.
2. Find the point where they cross.
3. Read its x- and y-coordinates.

Example: use the graph to solve y = x + 1 and x + y = 5.

1. Line A is y = x + 1. Rearranging x + y = 5 gives y = 5 − x, which is line B.
2. They cross at P(2, 3).
3. x = 2, y = 3

Check: 2 + 1 = 3 and 2 + 3 = 5. A graph gives exact answers only when the lines cross on a grid point, so check by substituting.

The roots of a quadratic equation = 0 are where its graph crosses the x-axis. To solve "quadratic = k", find where the curve crosses the horizontal line y = k.

Example: use the graph of y = x² − 4 (curve A) to solve x² − 4 = 0 and x² − 4 = 3.

1. Curve A crosses the x-axis at x = −2 and x = 2.
2. So x² − 4 = 0 gives x = −2 or x = 2
3. For x² − 4 = 3, find where the curve reaches a height of 3.
4. Reading the graph: x ≈ −2.6 or x ≈ 2.6

Check: x² = 7, so x = ±√7 = ±2.65 to 2 decimal places.

Where a line crosses a curve, both equations are true. The x-coordinates solve the equation you get by setting the two expressions equal.

1. Read the crossing points from the graph.
2. To show which equation they solve, set the two expressions equal and rearrange to = 0.

Example: the graph above shows y = x² − 4 (curve A) and y = x + 2 (line B). Find where they cross, and the equation this solves.

1. They cross at P(−2, 0) and Q(3, 5).
2. At the crossings, x² − 4 = x + 2
3. Rearrange: x² − x − 6 = 0
4. x = −2, y = 0 or x = 3, y = 5, which solve x² − x − 6 = 0

Check: x² − x − 6 = (x − 3)(x + 2), so the roots are 3 and −2.