Read the point of intersection of two lines

Which point lies on both graphed lines?

Line A stays y=x+1. Line B changes from y=5−x to y=7−x. Predict where their crossing moves

Move line B upwards and downwards. Follow the shared crossing rather than either axis intercept

As line B moves upwards, what happens to the crossing here?

Read one common point

For two nonparallel straight lines, one point belongs to both. Two different points, one from each line, do not form a solution

1. 1Horizontal first Trace from the crossing to the horizontal axis to read x.
2. 2Vertical second Use the same crossing to read y.
3. 3Check the grid scale A crossing can lie halfway between unit gridlines \(\displaystyle \frac{1+2}{2}=1.5\)
4. 4Follow the requested response Give both coordinates for a pair. If only x is requested, report just the horizontal coordinate

Read the horizontal coordinate of the crossing. Gridlines are one unit apart

The crossing lies halfway between the vertical gridlines at one and two

Read the vertical coordinate of the same crossing on this graph. Gridlines are one unit apart

Trace horizontally from the common point to the vertical axis

Which point is common to these two rising lines?

What is wrong with the student’s conclusion?

Read the common point of the two lines

Rule A gives 2+3x and rule B gives 7−x. Refine a trial x until both rules give the same output. The last column is A−B: zero means a shared point. Read x and both outputs together

Which pair gives equal outputs in both rules?

Model or data

\(\displaystyle 2+3x=7-x\)

Verify the refined crossing algebraically. Set the two outputs equal and find x.

Rule A gives 2+3x and rule B gives 7−x. If x counts whole items only, is there an allowed item count at which their outputs are equal?

Model or data

\(\displaystyle \begin{gathered}y=2x+1\\y=2x+4\end{gathered}\)

How many common points do these lines have?

Model or data

\(\displaystyle \begin{gathered}y=x+1\\2y=2x+2\end{gathered}\)

How many common points do these two equations describe?

Construct before reading

Use the same axes and scales for both equations. Plot at least two distinct points from each straight line, draw the lines, and read their common point

1. 1Calculate points Choose convenient inputs and calculate the corresponding outputs. Check each plotted point in its own equation
2. 2Use the full line A drawn window is not the whole coordinate plane. Extend the view if the equations predict a crossing outside it
3. 3Handle vertical lines For x=a, every point has the same horizontal coordinate. The line is vertical; it cannot be written as y=mx+b.
4. 4Confirm accuracy A plotted crossing gives an estimate at the displayed scale. Verify it in both original equations or refine numerically

Equations

\(\displaystyle y=2x-1\)

Which two points can be used to draw this line?

Equations

\(\displaystyle \begin{gathered}x=-2\\y=3\end{gathered}\)

Read the common point of the vertical and horizontal lines

Equations

\(\displaystyle \begin{gathered}y=x+1\\y=2x-19\end{gathered}\)

Only one line is visible in this window. Can you conclude that the two equations have no common point?

Equations

\(\displaystyle x+1=2x-19\)

Find the horizontal coordinate of the crossing by equating the outputs